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# Sequences determined by prime signature

This is a list of sequences in which membership is determined by their prime signature: if such a sequence contains a positive integer *n*, it contains all positive integers *n* with the same multiset of exponents in their prime factorizations.

## Contents

- 1 Properties
- 2 Examples
- 2.1 Sequences defined by number of divisors
- 2.2 Sequences defined by number of prime divisors with multiplicity
- 2.3 Sequences defined by number of prime divisors without multiplicity
- 2.4 Exponentially-S sequences
- 2.5 Sequences defined by minimum exponent
- 2.6 Sequences defined by maximum exponent
- 2.7 Sequences defined by GCD of exponent
- 2.8 Sequences defined by LCM of exponent
- 2.9 Sequences defined by product of exponent
- 2.10 Other sequences determined by prime signature
- 2.11 Uncategorized

- 3 Examples which need to be checked

## Properties

Apart from two trivial sequences (the empty sequence and the sequence containing only 1), these sequences have infinitely many members.

Any sequence where membership is determined only by the multiset of exponents, or functions based only on it, will be in this sequence. For example, sequences "Numbers n such that..." based on (only) the number of divisors, number of prime factors (distinct or with multiplicity), and so forth will necessarily have this property.

The smallest term of any such sequence which is greater than 1 must be even. If the sequence contains an odd term greater than 1, the number formed by replacing the least prime it its factorization by 2 is smaller.

If a sequence has this property, so does its (relative) complement in the natural numbers.

The set of integer sequences with this property is uncountably infinite and, in particular, has the same cardinality, as the set of all integer sequences. In particular this means that there is a bijection between integer sequences with this property and all integer sequences.

## Examples

Sequence A000027, the positive integers, trivially falls into all of the classes below and is omitted; the empty sequence is similar.

### Sequences defined by number of divisors

If , then either *n* and *m* are both in the sequence or neither are.

- A000040 The prime numbers: , signature (1)
- A001248 Squares of primes: , signature (2)
- A007624 Product of proper divisors of n = n^k, k>1: , signatures (1,1), (3), (2,1), (1,1,1), (3,1), (2,1,1), (1,1,1,1), (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), (1,1,1,1,1), ...
- A007964 Numbers n such that product of proper divisors of n is <= n: , signatures (), (1), (2), (1,1), (3)
- A009087 Numbers n such that the number of divisors of n is prime: signatures (1), (2), (4), (6), (10), (12), (16), (18), (22), (28), (30), (36), (40), (42), (46), (52), (58), (60), ...
- A018252 The nonprime numbers: , signatures (), (2), (1,1), (3), (2,1), (1,1,1), (4), (3,1), (2,2), (2,1,1), (1,1,1,1), (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), (1,1,1,1,1), ...
- A020725 Integers >= 2: (1), (2), (1,1), (3), (2,1), (1,1,1), (4), (3,1), (2,2), (2,1,1), (1,1,1,1), (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), (1,1,1,1,1), (6), ...
- A030513 Numbers with 4 divisors: (1,1), (3)
- A030514 4th powers of primes: (4)
- A030515 Numbers with exactly 6 divisors: (2,1), (5)
- A030516 Numbers with 7 divisors: (6)
- A030626 Numbers with 8 divisors: (1,1,1), (3,1), (7)
- A030627 Numbers with 9 divisors: (2,2), (8)
- A030628 1 together with numbers of the form p*q^4 and p^9: (), (4,1), (9)
- A030629 Numbers with 11 divisors: (10)
- A030630 Numbers with 12 divisors: (2,1,1), (3,2), (5,1), (11)
- A030631 Numbers with 13 divisors: (12)
- A030632 Numbers with 14 divisors: (6,1), (13)
- A030633 Numbers with 15 divisors: (4,2), (14)
- A030634 Numbers with 16 divisors: (1,1,1,1), (3,1,1), (3,3), (7,1), (15)
- A030635 Numbers with 17 divisors: (16)
- A030636 Numbers with 18 divisors: (2,2,1), (5,2), (8,1), (17)
- A030637 Numbers with 19 divisors: (18)
- A030638 Numbers with 20 divisors: (4,1,1), (4,3), (9,1), (19)
- A036436 Numbers n such that tau(n) is a square: (), (1,1), (3), (2,2), (1,1,1,1), (3,1,1), (3,3), (2,2,1,1), (1,1,1,1,1,1), (3,2,2), (3,1,1,1,1), (8), (7,1), ...
- A036455 Numbers n such that d(d(n)) is an odd prime: (1,1), (3), (2,2), (1,1,1,1), (3,1,1), (3,3), (1,1,1,1,1,1), (3,1,1,1,1), (8), (7,1), (4,4), (3,3,1,1), ...
- A036456 (2,1), (1,1,1), (3,1), (5), (4,1), (4,2), (2,2,2), (2,2,1,1), (7), (6,1), (3,2,2), (6,2), (5,2,1), (1,1,1,1,1,1,1,1), (9), (4,2,1,1,1), (2,2,2,1,1,1), ...
- A036457 (2,1,1), (3,2), (2,2,1), (2,1,1,1), (1,1,1,1,1), (5,1), (4,1,1), (3,2,1), (3,1,1,1), (2,1,1,1,1), (5,2), (4,3), (5,1,1), (4,2,1), (3,3,1), (4,1,1,1), ...
- A036458 (2,2,1,1,1), (2,1,1,1,1,1), (4,2,1,1), (3,2,2,1), (3,2,1,1,1), (2,2,2,1,1), (4,3,2), (5,2,1,1), (4,2,2,1), (3,3,2,1), (3,2,2,2), (5,1,1,1,1), (4,1,1,1,1,1), ...
- A036537 Numbers n such that number of divisors of n is a power of 2: (), (1), (1,1), (3), (1,1,1), (3,1), (1,1,1,1), (3,1,1), (1,1,1,1,1), (3,3), (3,1,1,1), ...
- A059269 Numbers n for which the number of divisors, tau(n), is divisible by 3: (2), (2,1), (2,2), (2,1,1), (5), (3,2), (2,2,1), (2,1,1,1), (5,1), (4,2), ...
- A063806 Numbers with a prime number of proper divisors: (2), (1,1), (3), (2,1), (1,1,1), (3,1), (2,1,1), (5), (3,2), (2,2,1), (2,1,1,1), (1,1,1,1,1), ...

### Sequences defined by number of prime divisors with multiplicity

If , then either *n* and *m* are both in the sequence or neither are.

- A000040 The prime numbers: , signature (1)
- A001358 Semiprimes (or biprimes): , signatures (2), (1,1)
- A002808 The composite numbers: , signatures (2), (1,1), (3), (2,1), (1,1,1), (4), (3,1), (2,2), (2,1,1), (1,1,1,1), (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), (1,1,1,1,1), ...
- A008578 Prime numbers at the beginning of the 20th century: , signatures (), (1)
- A014612 Numbers that are the product of exactly three (not necessarily distinct) primes: , signatures (3), (2,1), (1,1,1)
- A014613 Numbers that are products of 4 primes: , signatures (4), (3,1), (2,2), (2,1,1), (1,1,1,1)
- A014614 Numbers that are products of 5 primes: , signatures (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), (1,1,1,1,1)
- A018252 The nonprime numbers: , signatures (), (2), (1,1), (3), (2,1), (1,1,1), (4), (3,1), (2,2), (2,1,1), (1,1,1,1), (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), (1,1,1,1,1), ...
- A020725 Integers >= 2: (1), (2), (1,1), (3), (2,1), (1,1,1), (4), (3,1), (2,2), (2,1,1), (1,1,1,1), (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), (1,1,1,1,1), (6), ...
- A026424 Number of prime divisors (counted with multiplicity) is odd: (1), (3), (2,1), (1,1,1), (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), (1,1,1,1,1), (7), (6,1), ...
- A026478 a(n) = least positive integer > a(n-1) and not of form a(i)*a(j)*a(k) for 1<=i<=j<=k<n.
- A028260 Numbers n such that number of prime divisors of n (counted with multiplicity) is even: (), (2), (1,1), (4), (3,1), (2,2), (2,1,1), (1,1,1,1), (6), (5,1), (4,2), ...
- A028261 Numbers whose total number of prime factors (counting multiplicity) is squarefree: (1), (2), (1,1), (3), (2,1), (1,1,1), (5), (4,1), (3,2), (3,1,1), (2,2,1), ...
- A033942 At least 3 prime factors (counted with multiplicity): (3), (2,1), (1,1,1), (4), (3,1), (2,2), (2,1,1), (1,1,1,1), (5), (4,1), (3,2), (3,1,1), (2,2,1), ...
- A033987 Numbers that are divisible by at least 4 primes (counted with multiplicity): (4), (3,1), (2,2), (2,1,1), (1,1,1,1), (5), (4,1), (3,2), (3,1,1), (2,2,1), ...
- A037143 Numbers with at most 2 prime factors (counted with multiplicity): (), (1), (2), (1,1)
- A037144 Numbers with at most 3 prime factors (counted with multiplicity): (), (1), (2), (1,1), (3), (2,1), (1,1,1)
- A046304 Divisible by at least 5 primes (counted with multiplicity): (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), (1,1,1,1,1), (6), (5,1), (4,2), (3,3), (4,1,1), ...
- A046305 Divisible by at least 6 primes (counted with multiplicity): (6), (5,1), (4,2), (3,3), (4,1,1), (3,2,1), (2,2,2), (3,1,1,1), (2,2,1,1), (2,1,1,1,1), ...
- A046306 Numbers that are divisible by exactly 6 primes with multiplicity: (6), (5,1), (4,2), (3,3), (4,1,1), (3,2,1), (2,2,2), (3,1,1,1), (2,2,1,1), (2,1,1,1,1), (1,1,1,1,1,1)
- A046307 Numbers that are divisible by at least 7 primes (counted with multiplicity): (7), (6,1), (5,2), (4,3), (5,1,1), (4,2,1), (3,3,1), (3,2,2), (4,1,1,1), (3,2,1,1), ...
- A046308 Numbers that are divisible by exactly 7 primes counting multiplicity: (7), (6,1), (5,2), (4,3), (5,1,1), (4,2,1), (3,3,1), ..., (1,1,1,1,1,1,1)
- A046309 Numbers that are divisible by at least 8 primes (counted with multiplicity): (8), (7,1), (6,2), (5,3), (4,4), (6,1,1), (5,2,1), (4,3,1), (4,2,2), (3,3,2), ...
- A046310 Numbers that are divisible by exactly 8 primes counting multiplicity: (8), (7,1), (6,2), (5,3), (4,4), (6,1,1), (5,2,1), (4,3,1), (4,2,2), ..., (1,1,1,1,1,1,1,1)
- A046311 Numbers that are divisible by at least 9 primes (counted with multiplicity): (9), (8,1), (7,2), (6,3), (5,4), (7,1,1), (6,2,1), (5,3,1), (4,4,1), (5,2,2), ...
- A046312 Numbers that are divisible by exactly 9 primes with multiplicity: (9), (8,1), (7,2), (6,3), (5,4), (7,1,1), (6,2,1), (5,3,1), (4,4,1), ..., (1,1,1,1,1,1,1,1,1)
- A046313 Numbers that are divisible by at least 10 primes (counted with multiplicity): (10), (9,1), (8,2), (7,3), (6,4), (5,5), (8,1,1), (7,2,1), (6,3,1), (5,4,1), ...
- A046314 Numbers that are divisible by exactly 10 primes with multiplicity: (10), (9,1), (8,2), (7,3), (6,4), (5,5), (8,1,1), (7,2,1), ..., (1,1,1,1,1,1,1,1,1,1)
- A046339 Composite numbers with an odd number of prime factors (counted with multiplicity): (3), (2,1), (1,1,1), (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), ...
- A063989 Numbers with a prime number of prime divisors (counted with multiplicity): (2), (1,1), (3), (2,1), (1,1,1), (5), (4,1), (3,2), (3,1,1), (2,2,1), ...
- A065985 Numbers n such that d(n) / 2 is prime, where d(n) = number of divisors of n: (4), (2,2), (4,1), (3,2), (2,2,1), (6), (4,1,1), (3,2,1), (2,2,1,1), ...

### Sequences defined by number of prime divisors without multiplicity

If , then either *n* and *m* are both in the sequence or neither are.

- A000961 Powers of primes: , signatures (), (1), (2), (3), (4), (5), (6), (7), (8), (9), (10), (11), (12), (13), (14), (15), (16), (17), (18), (19), (20), (21), (22), (23), ...
- A000977 Numbers that are divisible by at least three different primes: , signatures (1,1,1), (2,1,1), (1,1,1,1), (3,1,1), (2,2,1), (2,1,1,1), (1,1,1,1,1), (4,1,1), (3,2,1), ...
- A007774 Numbers that are divisible by exactly 2 different primes: , signatures (1,1), (2,1), (3,1), (2,2), (4,1), (3,2), (5,1), (4,2), (3,3), (6,1), (5,2), (4,3), (7,1), (6,2), (5,3), ...
- A020725 Integers >= 2: (1), (2), (1,1), (3), (2,1), (1,1,1), (4), (3,1), (2,2), (2,1,1), (1,1,1,1), (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), (1,1,1,1,1), (6), ...
- A030230 Numbers n such that number of distinct primes dividing n is odd: (1), (2), (3), (1,1,1), (4), (2,1,1), (5), (3,1,1), (2,2,1), (1,1,1,1,1), (6), (4,1,1), ...
- A030231 Number of distinct primes dividing n is even: (), (1,1), (2,1), (3,1), (2,2), (1,1,1,1), (4,1), (3,2), (2,1,1,1), (5,1), (4,2), (3,3), (3,1,1,1), (2,2,1,1), ...
- A033992 Numbers that are divisible by exactly three different primes: (1,1,1), (2,1,1), (3,1,1), (2,2,1), (4,1,1), (3,2,1), (2,2,2), (5,1,1), (4,2,1), (3,3,1), ...
- A033993 Numbers that are divisible by exactly four different primes: (1,1,1,1), (2,1,1,1), (3,1,1,1), (2,2,1,1), (4,1,1,1), (3,2,1,1), (2,2,2,1), (5,1,1,1), ...
- A036116 Numbers n such that the number of distinct primes dividing n is a square: (), (1), (2), (3), (4), (1,1,1,1), (5), (2,1,1,1), (6), (3,1,1,1), (2,2,1,1), ...
- A051270 Numbers that are divisible by exactly 5 different primes: (1,1,1,1,1), (2,1,1,1,1), (3,1,1,1,1), (2,2,1,1,1), (4,1,1,1,1), (3,2,1,1,1), (2,2,2,1,1), ...
- A064040 Number of distinct prime divisors of n is a prime: (1,1), (2,1), (1,1,1), (3,1), (2,2), (2,1,1), (4,1), (3,2), (3,1,1), (2,2,1), (1,1,1,1,1), ...

### Exponentially-S sequences

A number is in such a sequence if and only if its prime exponents are a subset of *S*. If and are in the sequence, , and *p* and *q* are prime, then is in the sequence.

- A000452 The greedy sequence of integers which avoids 3-term geometric progressions: exponents are in A005836, signatures (), (1), (1,1), (3), (1,1,1), (4), (3,1), (1,1,1,1), (4,1), (3,1,1), (1,1,1,1,1), ...
- A001694 Powerful numbers: exponents are in A020725, signatures (), (2), (3), (4), (2,2), (5), (3,2), (6), (4,2), (3,3), (2,2,2), (7), (5,2), (4,3), (3,2,2), (8), (6,2), (5,3), (4,4), (4,2,2), (3,3,2), ...
- A002035 Numbers that contain primes to odd powers only: exponents are in A005408, signatures (), (1), (1,1), (3), (1,1,1), (3,1), (1,1,1,1), (5), (3,1,1), (1,1,1,1,1), (5,1), (3,3), (3,1,1,1), ...

### Sequences defined by minimum exponent

Numbers are included or excluded on the basis of the smallest of the (multi)set of their prime exponents. To avoid issues with min({}), 1 may be included or excluded without regard to other numbers.

- A001694 Powerful numbers: , signatures (), (2), (3), (4), (2,2), (5), (3,2), (6), (4,2), (3,3), (2,2,2), (7), (5,2), (4,3), (3,2,2), (8), (6,2), (5,3), (4,4), (4,2,2), (3,3,2), ...
- A036966 3-full (or cube-full, or cubefull) numbers: (), (3), (4), (5), (6), (3,3), (7), (4,3), (8), (5,3), (4,4), (9), (6,3), (5,4), (3,3,3), (10), (7,3), (6,4), ...
- A036967 4-full numbers: (), (4), (5), (6), (7), (8), (4,4), (9), (5,4), (10), (6,4), (5,5), (11), (7,4), (6,5), (12), (8,4), (7,5), (6,6), (4,4,4), (13), (9,4), ...
- A052485 Weak numbers: (1), (1,1), (2,1), (1,1,1), (3,1), (2,1,1), (1,1,1,1), (4,1), (3,1,1), (2,2,1), (2,1,1,1), (1,1,1,1,1), (5,1), (4,1,1), (3,2,1), ...

### Sequences defined by maximum exponent

Numbers are included or excluded on the basis of the largest of the (multi)set of their prime exponents. To avoid issues with max({}), 1 may be included or excluded without regard to other numbers.

- A004709 Cubefree numbers: , signatures (), (1), (2), (1,1), (2,1), (1,1,1), (2,2), (2,1,1), (1,1,1,1), (2,2,1), (2,1,1,1), (1,1,1,1,1), (2,2,2), (2,2,1,1), (2,1,1,1,1), ...
- A005117 Squarefree numbers: , signatures (1), (1,1), (1,1,1), (1,1,1,1), (1,1,1,1,1), (1,1,1,1,1,1), (1,1,1,1,1,1,1), (1,1,1,1,1,1,1,1), (1,1,1,1,1,1,1,1,1), (1,1,1,1,1,1,1,1,1,1), ...
- A013929 Numbers that are not squarefree: , signatures (2), (3), (2,1), (4), (3,1), (2,2), (2,1,1), (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), (6), (5,1), (4,2), (3,3), (4,1,1), ...
- A038109 Divisible exactly by the square of a prime: (2), (2,1), (2,2), (2,1,1), (3,2), (2,2,1), (2,1,1,1), (4,2), (3,2,1), (2,2,2), (2,2,1,1), (2,1,1,1,1), (5,2), ...
- A046099 Numbers that are not cubefree: (3), (4), (3,1), (5), (4,1), (3,2), (3,1,1), (6), (5,1), (4,2), (3,3), (4,1,1), (3,2,1), (3,1,1,1), (7), (6,1), (5,2), ...
- A046100 Biquadratefree numbers: (), (1), (2), (1,1), (3), (2,1), (1,1,1), (3,1), (2,2), (2,1,1), (1,1,1,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), (1,1,1,1,1), ...
- A046101 Biquadrateful numbers: (4), (5), (4,1), (6), (5,1), (4,2), (4,1,1), (7), (6,1), (5,2), (4,3), (5,1,1), (4,2,1), (4,1,1,1), (8), (7,1), (6,2), (5,3), (4,4), ...
- A060476 (), (3), (3,1), (5), (3,2), (3,1,1), (5,1), (3,3), (3,2,1), (3,1,1,1), (7), (5,2), (5,1,1), (3,3,1), (3,2,2), (3,2,1,1), (3,1,1,1,1), (8), (7,1), ...

### Sequences defined by GCD of exponent

Numbers are included or excluded on the basis of the GCD of the (multi)set of their prime exponents. To avoid issues with gcd({}), 1 may be included or excluded without regard to other numbers.

- A000037 Numbers that are not squares: , signatures (1), (1,1), (3), (2,1), (1,1,1), (3,1), (2,1,1), (1,1,1,1), (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), (1,1,1,1,1), ...
- A001597 Perfect powers: , signatures (), (2), (3), (4), (2,2), (5), (6), (4,2), (3,3), (2,2,2), (7), (8), (6,2), (4,4), (4,2,2), (2,2,2,2), (9), (6,3), (3,3,3), (10), (8,2), ...
- A007412 The noncubes: , signatures (1), (2), (1,1), (2,1), (1,1,1), (4), (3,1), (2,2), (2,1,1), (1,1,1,1), (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), (1,1,1,1,1), (5,1), (4,2), ...
- A007916 Not a perfect power: , signatures (1), (1,1), (2,1), (1,1,1), (3,1), (2,1,1), (1,1,1,1), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), (1,1,1,1,1), (5,1), (4,1,1), (3,2,1), ...
- A056798 Prime powers with even exponents >=0: (), (2), (4), (6), (8), (10), (12), (14), (16), (18), (20), (22), (24), (26), (28), (30), (32), (34), (36), ...

### Sequences defined by LCM of exponent

Numbers are included or excluded on the basis of the LCM of the (multi)set of their prime exponents. To avoid issues with lcm({}), 1 may be included or excluded without regard to other numbers.

- A002035 Numbers that contain primes to odd powers only: , signatures (), (1), (1,1), (3), (1,1,1), (3,1), (1,1,1,1), (5), (3,1,1), (1,1,1,1,1), (5,1), (3,3), (3,1,1,1), ...

### Sequences defined by product of exponent

- A048107 Number of unitary divisors of n (A034444) > number of non-unitary divisors of n (A048105): (), (1), (2), (1,1), (2,1), (1,1,1), (2,1,1), (1,1,1,1), ...
- A048108 Numbers n with at least as many non-unitary divisors as unitary divisors: (3), (4), (3,1), (2,2), (5), (4,1), (3,2), (3,1,1), (2,2,1), (6), (5,1), ...
- A048109 (3), (3,1), (3,1,1), (3,1,1,1), (3,1,1,1,1), (3,1,1,1,1,1), (3,1,1,1,1,1,1), (3,1,1,1,1,1,1,1), (3,1,1,1,1,1,1,1,1), (3,1,1,1,1,1,1,1,1,1), (3,1,1,1,1,1,1,1,1,1,1), ...
- A048111 (4), (2,2), (5), (4,1), (3,2), (2,2,1), (6), (5,1), (4,2), (3,3), (4,1,1), (3,2,1), (2,2,2), (2,2,1,1), (7), (6,1), (5,2), (4,3), (5,1,1), (4,2,1), (3,3,1), ...
- A060687 Numbers n such that there exist exactly 2 Abelian groups of order n: (2), (2,1), (2,1,1), (2,1,1,1), (2,1,1,1,1), (2,1,1,1,1,1), (2,1,1,1,1,1,1), ...

### Other sequences determined by prime signature

- A000028 (1), (2), (1,1,1), (4), (3,1), (2,1,1), (3,2), (2,2,1), (1,1,1,1,1), (5,1), (4,1,1), (2,2,2), (3,1,1,1), (2,1,1,1,1), (7), (6,1), (5,2), (4,3), (4,2,1), ...
- A000379 (), (1,1), (3), (2,1), (2,2), (1,1,1,1), (5), (4,1), (3,1,1), (2,1,1,1), (6), (4,2), (3,3), (3,2,1), (2,2,1,1), (1,1,1,1,1,1), (5,1,1), (3,2,2), (4,1,1,1), ...
- A000430 Primes and squares of primes: (1), (2)
- A000469 1 together with products of 2 or more distinct primes: (), (1,1), (1,1,1), (1,1,1,1), (1,1,1,1,1), (1,1,1,1,1,1), (1,1,1,1,1,1,1), (1,1,1,1,1,1,1,1), ...
- A006881 Squarefree semiprimes: (1,1)
- A007304 Sphenic numbers: (1,1,1)
- A007422 Multiplicatively perfect numbers n: (), (1,1), (3)
- A024619 Numbers that are not powers of primes p^k (k >= 0): (1,1), (2,1), (1,1,1), (3,1), (2,2), (2,1,1), (1,1,1,1), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), ...
- A025475 1 and the prime powers p^m where m >= 2, thus excluding the primes: (), (2), (3), (4), (5), (6), (7), (8), (9), (10), (11), (12), (13), (14), (15), (16), ...
- A026416 (), (1), (2), (1,1,1), (4), (3,1), (2,1,1), (3,2), (2,2,1), (1,1,1,1,1), (5,1), (4,1,1), (2,2,2), (3,1,1,1), (2,1,1,1,1), (7), (6,1), (5,2), (4,3), (4,2,1), ...
- A026422 (), (1), (3), (2,1), (1,1,1), (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), (1,1,1,1,1), (7), (6,1), (5,2), (4,3), (5,1,1), (4,2,1), (3,3,1), (3,2,2), ...
- A026477 (), (1), (2), (4), (1,1,1,1), (3,1,1), (3,3), (2,2,2,1), (1,1,1,1,1,1,1), (8), (7,1), (3,1,1,1,1,1), (6,2,1), (5,2,2), (3,3,1,1,1), (6,2,1), (5,2,2), ...
- A030059 Numbers that are the product of an odd number of distinct primes: (1), (1,1,1), (1,1,1,1,1), (1,1,1,1,1,1,1), (1,1,1,1,1,1,1,1,1), (1,1,1,1,1,1,1,1,1,1,1), ...
- A030078 Cubes of primes: (3)
- A030140 The nonsquares squared: (2), (2,2), (6), (4,2), (2,2,2), (6,2), (4,2,2), (2,2,2,2), (10), (8,2), (6,4), (6,2,2), (4,4,2), (4,2,2,2), (2,2,2,2,2), (10,2), ...
- A030229 Product of an even number of distinct primes: (), (1,1), (1,1,1,1), (1,1,1,1,1,1), (1,1,1,1,1,1,1,1), (1,1,1,1,1,1,1,1,1,1), (1,1,1,1,1,1,1,1,1,1,1,1), ...
- A036454 Prime powers with special exponents: q^(p-1) where p > 2: (2), (4), (6), (10), (12), (16), (18), (22), (28), (30), (36), (40), (42), (46), (52), (58), ...
- A036785 Numbers divisible by the squares of two distinct primes: (2,2), (3,2), (2,2,1), (4,2), (3,3), (3,2,1), (2,2,2), (2,2,1,1), (5,2), (4,3), (4,2,1), (3,3,1), ...
- A046386 Products of four distinct primes: (1,1,1,1)
- A046387 Products of 5 distinct primes: (1,1,1,1,1)
- A048943 Product of divisors of n is a square: (), (1,1), (3), (1,1,1), (4), (3,1), (2,1,1), (1,1,1,1), (3,2), (3,1,1), (2,1,1,1), (1,1,1,1,1), (5,1), (3,3), (4,1,1), ...
- A048944 Product of divisors of n is a cube: (), (2), (3), (2,1), (2,2), (2,1,1), (5), (3,2), (2,2,1), (2,1,1,1), (6), (5,1), (4,2), (3,3), (3,2,1), (2,2,2), (2,2,1,1), ...
- A048945 Numbers n such that product of divisors of n is a fourth power: (), (1,1,1), (3,1), (1,1,1,1), (3,1,1), (2,1,1,1), (1,1,1,1,1), (3,3), (3,2,1), ...
- A048946 Product of divisors of n is a fifth power: (), (4), (5), (4,1), (4,2), (4,1,1), (4,3), (4,2,1), (4,1,1,1), (4,4), (4,3,1), (4,2,2), (4,2,1,1), ...
- A050376 Numbers of the form p^(2^k) where p is prime and k >= 0: (1), (2), (4), (8), (16), (32), (64), (128), (256), (512), (1024), (2048), (4096), (8192), ...
- A051144 Nonsquarefree nonsquares: (3), (2,1), (3,1), (2,1,1), (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), (5,1), (3,3), (4,1,1), (3,2,1), (3,1,1,1), ...
- A051676 Composite numbers n such that the number of divisors of n^2 is a prime: (2), (3), (5), (6), (8), (9), (11), (14), (15), (18), (20), (21), (23), ...
- A052486 Achilles numbers: (3,2), (5,2), (4,3), (3,2,2), (5,3), (3,3,2), (7,2), (5,4), (5,2,2), (4,3,2), (3,2,2,2), (7,3), (5,3,2), (4,3,3), (3,3,2,2), ...
- A053810 Prime powers of prime numbers: (2), (3), (5), (7), (11), (13), (17), (19), (23), (29), (31), (37), (41), (43), (47), (53), (59), (61), (67), (71), ...
- A054753 Numbers which are the product of a prime and the square of a different prime: (2,1)
- A056166 n is the product of distinct primes raised to prime powers: (), (2), (3), (2,2), (5), (3,2), (3,3), (2,2,2), (7), (5,2), (3,2,2), (5,3), (3,3,2), ...
- A056824 Odd powers of a prime but not prime: (3), (5), (7), (9), (11), (13), (15), (17), (19), (21), (23), (25), (27), (29), (31), (33), (35), (37), (39), (41), ...
- A058080 Numbers n such that product of divisors of n is > n^2: (2,1), (1,1,1), (4), (3,1), (2,2), (2,1,1), (1,1,1,1), (5), (4,1), (3,2), (3,1,1), (2,2,1), ...
- A059404 Numbers n such that n/[largest power of squarefree kernel] is larger than 1: (2,1), (3,1), (2,1,1), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), ...
- A062171 (3), (4), (3,1), (2,2), (2,1,1), (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), (6), (5,1), (4,2), (3,3), (4,1,1), (3,2,1), (2,2,2), (3,1,1,1), ...
- A062312 Nonprime numbers squared: (), (4), (2,2), (6), (4,2), (2,2,2), (8), (6,2), (4,4), (4,2,2), (2,2,2,2), (10), (8,2), (6,4), (6,2,2), (4,4,2), ...
- A062320 Nonsquarefree numbers squared: (4), (6), (4,2), (8), (6,2), (4,4), (4,2,2), (10), (8,2), (6,4), (6,2,2), (4,4,2), (4,2,2,2), (12), (10,2), (8,4), ...
- A062503 Squarefree numbers squared: (), (2), (2,2), (2,2,2), (2,2,2,2), (2,2,2,2,2), (2,2,2,2,2,2), (2,2,2,2,2,2,2), (2,2,2,2,2,2,2,2), (2,2,2,2,2,2,2,2,2), ...
- A062770 n/[largest power of squarefree kernel] equals 1: (), (1), (2), (1,1), (3), (1,1,1), (4), (2,2), (1,1,1,1), (5), (1,1,1,1,1), (6), (3,3), (2,2,2), ...
- A062838 Cubes of squarefree numbers: (), (3), (3,3), (3,3,3), (3,3,3,3), (3,3,3,3,3), (3,3,3,3,3,3), (3,3,3,3,3,3,3), (3,3,3,3,3,3,3,3), (3,3,3,3,3,3,3,3,3), ...
- A063774 Number of divisors of n^2 is a square: (), (1,1), (4), (2,2), (1,1,1,1), (3,3), (4,1,1), (2,2,1,1), (1,1,1,1,1,1), (4,4), (4,2,2), (3,3,1,1), ...
- A064499 Composite numbers n such that product of aliquot divisors of n is a perfect square: (2,1), (4), (5), (4,1), (2,2,1), (6,1), (5,2), (4,2,1), ...
- A065036 Product of the cube of a prime (A030078) and a different prime: (3,2)
- A065127 Nonsquares with number of prime factors equal to twice the number of distinct prime factors: (), (2), (3,1), (2,2), (4,1,1), (3,2,1), (2,2,2), ...

### Uncategorized

- A066423 Product of proper divisors of the n-th composite number does not equal the n-th composite number: (2), (2,1), (1,1,1), (4), (3,1), (2,2), (2,1,1), ...
- A066427 Numbers with mu = 0 and infinitary MoebiusMu = -1; (sum of binary digits of prime exponents is odd): (2), (4), (3,1), (2,1,1), (3,2), (2,2,1), ...
- A066428 Numbers with mu = 0 and infinitary MoebiusMu = +1 (sum of binary digits of prime exponents is even): (3), (2,1), (2,2), (5), (4,1), (3,1,1), ...
- A067028 Numbers with a composite number of prime factors (counted with multiplicity): (4), (3,1), (2,2), (2,1,1), (1,1,1,1), (6), (5,1), (4,2), (3,3), ...
- A067259 Cubefree numbers which are not squarefree: (2), (2,1), (2,2), (2,1,1), (2,2,1), (2,1,1,1), (2,2,2), (2,2,1,1), (2,1,1,1,1), (2,2,2,1), (2,2,1,1,1), ...
- A067340 Number of prime factors divided by the number of distinct prime factors is an integer: (1), (2), (1,1), (3), (1,1,1), (4), (3,1), (2,2), (1,1,1,1), ...
- A067341 Number of prime factors divided by the number of distinct prime factors is an integer and n is neither squarefree, nor power of prime: (3,1), (2,2), (5,1), ...
- A067582 Nonprime values of n such that bigomega(n)^omega(n)=omega(n)^bigomega(n): (), (1,1), (1,1,1), (3,1), (2,2), (1,1,1,1), (1,1,1,1,1), (1,1,1,1,1,1), ...
- A067801 Numbers n such that bigomega(n)=2*omega(n): (), (2), (3,1), (2,2), (4,1,1), (3,2,1), (2,2,2), (5,1,1,1), (4,2,1,1), (3,3,1,1), (3,2,2,1), (2,2,2,2), ...
- A068993 Numbers n such that A062799(n)=4: (1,1), (4)
- A069272 11-almost primes (generalization of semiprimes): (11), (10,1), (9,2), (8,3), (7,4), (6,5), (9,1,1), (8,2,1), (7,3,1), (6,4,1), (5,5,1), (7,2,2), ...
- A069781 n is such that GCD[d(n^3),d(n)] is not a power of 2: (4,3), (6,2), (4,3,1), (6,2,1), (4,3,2), (4,3,1,1), (6,2,2), (4,3,3), (6,2,1,1), (4,3,2,1), ...
- A069782 Numbers n such that GCD(d(n^3),d(n)) = 2^w for some w: (), (1), (2), (1,1), (3), (2,1), (1,1,1), (4), (3,1), (2,2), (2,1,1), (1,1,1,1), (5), (4,1), ...
- A070011 Numbers n such that number of prime factors divided by the number of distinct prime factors is not an integer: (2,1), (2,1,1), (4,1), (3,2), ...
- A070265 Odd powers: (), (3), (5), (6), (3,3), (7), (9), (6,3), (3,3,3), (10), (5,5), (11), (12), (9,3), (6,6), (6,3,3), (3,3,3,3), (13), (14), (7,7), (15), ...
- A070915 Numbers having at most two distinct prime factors: (), (1), (2), (1,1), (3), (2,1), (4), (3,1), (2,2), (5), (4,1), (3,2), (6), (5,1), (4,2), (3,3), ...
- A072357 Cube-free nonsquares having exactly one square in their factorization: (2,1), (2,1,1), (2,1,1,1), (2,1,1,1,1), (2,1,1,1,1,1), (2,1,1,1,1,1,1), ...
- A072412 LCM of exponents in prime factorization of n does not equal the largest exponent: (3,2), (3,2,1), (5,2), (4,3), (3,2,2), (3,2,1,1), (5,3), (5,2,1), ...
- A072413 LCM of exponents in prime factorization of n does not equal the product of the exponents: (2,2), (3,2), (2,2,1), (4,2), (3,3), (3,2,1), (2,2,2), ...
- A072414 Non-Achilles numbers for which LCM of the exponents in the prime factorization of n is not equal to the maximum of the same exponents: (3,2,1), (3,2,1,1), ...
- A072587 Numbers having at least one prime factor with an even exponent: (2), (2,1), (4), (2,2), (2,1,1), (4,1), (3,2), (2,2,1), (2,1,1,1), (6), (4,2), ...
- A072588 Numbers having at least one prime factor with an odd and one with an even exponent: (2,1), (2,1,1), (4,1), (3,2), (2,2,1), (2,1,1,1), (4,1,1), ...
- A072774 Powers of squarefree numbers: (), (1), (2), (1,1), (3), (1,1,1), (4), (2,2), (1,1,1,1), (5), (1,1,1,1,1), (6), (3,3), (2,2,2), (1,1,1,1,1,1), (7), ...
- A072777 Powers of squarefree numbers which are not squarefree: (2), (3), (4), (2,2), (5), (6), (3,3), (2,2,2), (7), (8), (4,4), (2,2,2,2), (9), (3,3,3), ...
- A074451 Non-cubefree noncubes: (4), (3,1), (5), (4,1), (3,2), (3,1,1), (5,1), (4,2), (4,1,1), (3,2,1), (3,1,1,1), (7), (6,1), (5,2), (4,3), (5,1,1), ...
- A074661 Numbers n such that max{e_2, e_3, ...} is prime: (2), (3), (2,1), (3,1), (2,2), (2,1,1), (5), (3,2), (3,1,1), (2,2,1), (2,1,1,1), (5,1), (3,3), ...
- A074853 Numbers n not in A065036 but such that tau(n)=omega(n)^3: (2,2,2), (3,3,1,1), (2,2,2,1,1,1), (7,1,1,1), (3,1,1,1,1,1,1,1)
- A074985 Squares of semiprimes: (4), (2,2)
- A076292 Perfect powers with squarefree root: (2), (3), (4), (2,2), (5), (6), (3,3), (2,2,2), (7), (8), (4,4), (2,2,2,2), (9), (3,3,3), (10), (5,5), ...
- A076467 Perfect powers m^k where m is an integer and k > 2: (3), (4), (5), (6), (3,3), (7), (8), (4,4), (9), (6,3), (3,3,3), (10), (5,5), (11), (12), (9,3), ...
- A077438 Numbers n such that sum( d|n : mu(d)mu(n/d)^2) = -1: (2), (2,2,2), (2,2,2,2,2), (2,2,2,2,2,2,2), (2,2,2,2,2,2,2,2,2), (2,2,2,2,2,2,2,2,2,2,2), ...
- A077448 Numbers n such that sum( d|n : mu(d)mu(n/d)^2) = +1: (), (2,2), (2,2,2,2), (2,2,2,2,2,2), (2,2,2,2,2,2,2,2), (2,2,2,2,2,2,2,2,2,2), ...
- A079712 Numbers n such that bigomega(n) = 3*omega(n): (), (3), (5,1), (4,2), (3,3), (7,1,1), (6,2,1), (5,3,1), (4,4,1), (5,2,2), (4,3,2), (3,3,3), ...
- A080257 Numbers having at least two distinct or a total of at least three prime factors: (1,1), (3), (2,1), (1,1,1), (4), (3,1), (2,2), (2,1,1), (1,1,1,1), (5), ...
- A080258 Either 4th power of a prime, or product of a prime and the square of a different prime: (2,1), (4)
- A081619 Numbers whose divisors can be arranged as equilateral triangle: (), (2), (2,1), (5), (4,1), (4,2), (2,2,1,1), (3,2,2), (6,2), (6,1,1), (5,2,1), ...
- A082293 Numbers having exactly one square divisor > 1: (2), (3), (2,1), (3,1), (2,1,1), (3,1,1), (2,1,1,1), (3,1,1,1), (2,1,1,1,1), (3,1,1,1,1), ...
- A082294 Numbers having exactly two square divisors > 1: (4), (5), (4,1), (5,1), (4,1,1), (5,1,1), (4,1,1,1), (5,1,1,1), (4,1,1,1,1), (5,1,1,1,1), ...
- A082295 Numbers having more than two square divisors > 1: (2,2), (3,2), (2,2,1), (6), (4,2), (3,3), (3,2,1), (2,2,2), (2,2,1,1), (7), (6,1), (5,2), (4,3), ...
- A082522 p^(2^k) with p prime and k>0: (2), (4), (8), (16), (32), (64), (128), (256), (512), (1024), (2048), (4096), (8192), (16384), (32768), (65536), ...
- A084116 Numbers m such that A084115(m) = 1: (1), (1,1), (3)
- A084227 Numbers of the form p*q^k with distinct primes p and q, k>0: (1,1), (2,1), (3,1), (4,1), (5,1), (6,1), (7,1), (8,1), (9,1), (10,1), (11,1), (12,1), ...
- A084384 a(1) = 2; a(n+1) = smallest k > a(n) that is divisible by at most (1/2)*[tau(k)] previous terms: (1), (2), (1,1), (3), (5), (4,1), (3,2), (3,1,1), ...
- A084400 a(1) = 1; for n>1, a(n) = smallest number that does not divide the product of all previous terms: (), (1), (2), (4), (8), (16), (32), (64), (128), ...
- A084679 Composite numbers with coprime numbers of prime factors with and without repetition: (2,1), (2,1,1), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), ...
- A085155 Powers of semiprimes: (), (2), (1,1), (4), (2,2), (6), (3,3), (8), (4,4), (10), (5,5), (12), (6,6), (14), (7,7), (16), (8,8), (18), (9,9), (20), ...
- A085156 Powers of primes or of semiprimes: (), (1), (2), (1,1), (3), (4), (2,2), (5), (6), (3,3), (7), (8), (4,4), (9), (10), (5,5), (11), (12), (6,6), ...
- A085971 Union of primes and numbers that are not prime powers (A000040, A024619): (1), (1,1), (2,1), (1,1,1), (3,1), (2,2), (2,1,1), (1,1,1,1), (4,1), (3,2), ...
- A085986 Squares of the squarefree semiprimes: (2,2)
- A085987 Product of exactly four primes, three of which are distinct: (2,1,1)
- A087797 Primes, squares of primes and cubes of primes: (1), (2), (3)
- A089229 Neither primes nor square numbers: (1,1), (3), (2,1), (1,1,1), (3,1), (2,1,1), (1,1,1,1), (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), ...
- A093599 Composite numbers having an odd number of prime factors, all of which are distinct: (1,1,1), (1,1,1,1,1), (1,1,1,1,1,1,1), (1,1,1,1,1,1,1,1,1), ...
- A093771 Perfect powers for which the exponent is a prime number: solutions to {A051409(x) is prime}: (2), (3), (2,2), (5), (4,2), (3,3), (2,2,2), (7), ...
- A094784 Numbers that are neither squares nor cubes: (1), (1,1), (2,1), (1,1,1), (3,1), (2,1,1), (1,1,1,1), (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), ...
- A096165 Prime powers with exponents that are themselves prime powers: (1), (2), (3), (4), (5), (7), (8), (9), (11), (13), (16), (17), (19), (23), (29), ...
- A096432 Numbers n such that 1 + max{e_2, e_3, ...} is a prime: (1), (2), (1,1), (2,1), (1,1,1), (4), (2,2), (2,1,1), (1,1,1,1), (4,1), (2,2,1), (2,1,1,1), ...
- A100959 Non-semiprimes: (), (1), (3), (2,1), (1,1,1), (4), (3,1), (2,2), (2,1,1), (1,1,1,1), (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), (1,1,1,1,1), ...
- A102466 Numbers such that the number of divisors is the sum of numbers of prime factors with and without repetitions: (1), (2), (1,1), (3), (4), (5), (6), ...
- A102467 Numbers such that the sum of numbers of prime factors with and without repetitions does not equal the number of divisors: (), (2,1), (1,1,1), (3,1), ...
- A102562 Numbers n such that the cyclic group of order n is not a Hajós group: (3,2), (3,1,1), (2,2,1), (2,1,1,1), (1,1,1,1,1), (4,2), (3,3), (4,1,1), (3,2,1), ...
- A102834 Numbers whose factors are primes raised to powers >= 2 and are not perfect squares: (3), (5), (3,2), (3,3), (7), (5,2), (4,3), (3,2,2), (5,3), ...
- A105642 Composite nonsquares and noncubes: (1,1), (2,1), (1,1,1), (3,1), (2,1,1), (1,1,1,1), (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), (1,1,1,1,1), ...
- A106543 Composite numbers that are not perfect powers: (1,1), (2,1), (1,1,1), (3,1), (2,1,1), (1,1,1,1), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), ...
- A109399 Numbers with at least two 3s in their prime signature: (3,3), (3,3,1), (3,3,2), (3,3,1,1), (3,3,3), (3,3,2,1), (3,3,1,1,1), (4,3,3), (3,3,3,1), ...

## Examples which need to be checked

- A088480 Numbers n such that the lunar product of the distinct lunar prime divisors of n is >= n:
- A089105 Values taken by least witness function W(n):

- A109421 Numbers n such that tau(n)/bigomega(n) is an integer [tau(n)=number of divisors of n; bigomega(n)=number of prime divisors of n, counted with multiplicities]:
- A109422 Numbers n such that tau(n)/bigomega(n) is not an integer [tau(n) =number of divisors of n; bigomega(n)=number of prime divisors of n, counted with multiplicities]:
- A109425 Numbers n such that tau(n)/omega(n) is an integer [tau(n) =number of divisors of n; omega(n)=number of distinct prime factors of n]:
- A109426 Numbers n such that tau(n)/omega(n) is not an integer [tau(n) =number of divisors of n; omega(n)=number of distinct prime factors of n]:
- A110893 Numbers with a semiprime number of prime divisors (counted with multiplicity):
- A111030 Magic products of 4 X 4 multiplicative magic squares:
- A111087 Neither primes nor semiprimes:
- A111307 Numbers which are perfect powers m^k, where m is an integer, equal to the sum of m consecutive primes plus another prime which is larger than all the others:
- A111398 Numbers which are the cube roots of the product of their proper divisors:
- A111399 Numbers in A048945 but not in A111398:
- A114127 Numbers that factorize into a prime number of prime factors each raised to a prime exponent:
- A114128 Numbers that factorize into a prime number of distinct prime factors each raised to a different prime exponent:
- A114129 Numbers that factorize into a set of prime factors that are each raised to a different prime exponent:
- A114987 Numbers with a 3-almost prime number of prime divisors (counted with multiplicity):
- A115063 Natural numbers of the form p^F(n)*q^F(n)*r^F(n)*...*z^F(n), where p,q,r,... are distinct primes and F(n) is a Fibonacci number:
- A115105 Numbers of the form p^F(n)*q^F(n), where p and q are distinct primes; F(n) is a Fibonacci number:
- A115975 Numbers of the form p^k, where p is a prime and k is a Fibonacci number:
- A119251 Positive integers each with exactly 1 unitary prime divisor (i.e. n is included if and only if A056169(n) = 1):
- A119675 Natural numbers n such that the number of prime factors of n (counted with multiplicity) is a Fibonacci number:
- A119847 Positions where A119842 is zero:
- A119848 Positions where A119842 is not zero:
- A119850 Positions where A119842 is greater than one:
- A119885 Natural numbers with number of divisors equal to a Lucas number:
- A119899 Integers i such that bigomega(i) (A001222) and tau(i) (A000005) are both even:
- A119911 Natural numbers with number of divisors not equal to a Lucas number:
- A120497 Natural numbers n with number of divisors equal to a perfect power:
- A120944 Composite squarefree numbers:
- A122181 Numbers n that can be written as n = x*y*z with 1<x<y<z (A122180(n)>0):
- A123193 Natural numbers with number of divisors equal to a Fibonacci number: (), (1), (2), (1,1,1), (4), (3,1), (7), (6,2), (2,2,1,1,1,1), (3,2,2,1,1), (3,3,2,2), ...
- A123240 Natural numbers with number of divisors not equal to a Fibonacci number:
- A123711 Indices n such that A123709(n) = 8 = number of nonzero terms in row n of triangle A123706:
- A123712 Indices n such that 16 = A123709(n) = number of nonzero terms in row n of triangle A123706:
- A126706 Positive integers which are neither squarefree integers nor prime powers:
- A128603 Numbers dividing p^6 for p a prime:
- A130091 Numbers having in their canonical prime factorization mutually distinct exponents:
- A130092 Numbers with at least two factors having in their canonical prime factorization equal exponents:
- A130446 Integers in [1, 425] expressible as a difference of the terms of the unique optimal Golomb ruler of order 24. See A130444:
- A130763 Natural numbers such that d(n)!+ 1 is a square, where d(n) is the number of divisors of n, A000005:
- A130897 Numbers that are not exponentially squarefree:
- A131181 A 2-way classification of integers: complement of A026416:
- A131605 Perfect powers of nonprimes (m^k where m is a nonprime positive integer and k >= 2):
- A134612 Nonprime numbers such that the root mean cube of their prime factors is a prime (where the root mean cube of c and d is ((c^3+d^3)/2)^(1/3)):
- A137487 Numbers with 24 divisors:
- A137491 Numbers with 28 divisors:
- A137493 Numbers with 30 divisors:
- A137944 Numbers such that the number of composite divisors is a multiple of the number of prime divisors; a(1)=1:
- A137945 Non-prime-powers such that the number of composite divisors is a multiple of the number of prime divisors:
- A138302 Products of distinct relatively prime terms of A084400:
- A139118 Numbers with a nonprime number of divisors:
- A139588 Nonprime numbers with Fibonacci number of divisors:
- A140823 Natural numbers which are not perfect fourth powers:
- A143610 Numbers of the form p^2*q^3, where p,q are distinct primes:
- A144338 Squarefree numbers > 1:
- A144972 Power-6-free numbers:
- A145784 Numbers with property that the number of prime factors is a multiple of 3:
- A153158 A007916(n)^2:
- A154893 Numbers whose number of proper divisors is not a prime number:
- A158340 Composite numbers k such that number of prime factors of k + number of divisors of k = prime:
- A162643 Numbers such that their number of divisors is not a binary power:
- A162644 Numbers m such that A162511(m) = +1:
- A162645 Numbers m such that A162511(m) = -1:
- A162947 Numbers n such that the product of all divisors of n equals n^3:
- A162966 Union of 1 and nonsquarefree numbers (A013929):
- A163569 Numbers of the form p^3*q^2*r where p, q and r are three distinct primes:
- A164336 a(1)=1. Thereafter, all terms are primes raised to the values of earlier terms of the sequence:
- A164514 1 followed by numbers that are not squares:
- A166155 Numbers n such that number of divisors of n + number of perfect partitions of (n-1) is prime:
- A166546 Natural numbers n such that d(n) + 1 is prime:
- A166684 Numbers n such that d(n)<4:
- A166718 Numbers with at most 4 prime factors (counted with multiplicity):
- A166719 Numbers with at most 5 prime factors (counted with multiplicity):
- A166982 Natural numbers with number of perfect partitions equal to a perfect power:
- A167171 Numbers such that d(n)=2*omega(n), where d = A000005 is the number of divisors:
- A167175 Numbers with a nonprime number of prime divisors (counted with multiplicity):
- A167207 Numbers that are not divisible by a smaller number that is a square greater than 1:
- A167758 Numbers n such that d(n)=nonisolated nonprime:
- A167759 Numbers n such that d(n)=isolated number:
- A168086 Numbers n such that d(n)=nonisolated number:
- A168363 Squares and cubes of primes:
- A168638 Number of distinct prime divisors of n is 2 or 3:
- A168645 Numbers with 2 or 3 prime divisors (counted with multiplicity):
- A171474 n-th prime^n-th nonnegative nonprime (without repetition):
- A171561 (n-th prime number)^(n-th non-single or nonisolated number) with duplicates removed:
- A172443 Numbers with exactly 64 divisors:
- A173743 Numbers n such that phi(tau(n))= tau(rad(n)) :
- A174891 Row indices for nonzero elements in first column of A174888:
- A174895 a(n) = possible values of A007955(m) in increasing order, where A007955(m) = product of divisors of m:
- A174896 a(n) = numbers k in increasing order such that A007955(m) = k has no solution for any m, where A007955(m) = product of divisors of m:
- A175050 Positive integers n where both n and the number of divisors of n are perfect powers. (Both n and d(n) are elements of A001597.):
- A175082 Possible values for sum of perfect divisors of n:
- A175084 Possible values for product of perfect divisors of n:
- A175085 Numbers m such that product of perfect divisors of x = m has no solution:
- A175086 Perfect powers m such that product of perfect divisors of x = m has solution:
- A175391 Perfect squares each with a square number of divisors:
- A175496 Those positive integers n where n is not a squarefree integer but the number of divisors of n is a power of 2:
- A175742 Numbers with 32 divisors:
- A175746 Numbers with 36 divisors:
- A175749 Numbers with 40 divisors:
- A175750 Numbers with 42 divisors:
- A175754 Numbers with 48 divisors:
- A176238 Natural numbers n such that d(d(n)+1) > 2:
- A176297 Numbers with at least one 3 in their prime signature:
- A176525 Fermi-Dirac semiprimes: products of two distinct terms of A050376:
- A176540 1 together with the semiprimes:
- A177425 Integers with multiple and strictly distinct powers:
- A177492 Products of squares of 2 or more distinct primes:
- A177880 Numbers such that not all exponents in prime power factorization are in A005836:
- A177899 Nonsquarefree numbers that are not in A177880:
- A178212 Nonsquarefree numbers divisible by exactly three distinct primes:
- A178739 Product of the 4th power of a prime (A030514) and a different prime:
- A178740 Product of the 5th power of a prime (A050997) and a different prime:
- A179126 Positive integers n for which the torsion subgroup of the elliptic curve y^2 = x^3 + n has order 3:
- A179642 Product of exactly 5 primes, 3 of which are distinct:
- A179643 Products of exactly 2 distinct primes squares and a different prime (p^2*q^2*r):
- A179644 Product of the 4th power of a prime and 2 different distinct primes (p^4*q*r):
- A179664 Products of the 7th power of a prime and a distinct prime (p^7*q):
- A179667 Products of the 5th power of a prime and 2 distinct primes (p^5*q*r):
- A179668 Products of the 8th power of a prime and a distinct prime (p^8*q):
- A179669 Products of form p^4*q^2*r where p, q and r are three distinct primes:
- A179670 Products of the 3rd power of a prime and 3 distinct primes (p^3*q*r*s):
- A179672 Products of the 6st power of a prime and 2 distinct primes (p^6*q*r):
- A179688 Numbers of the form p^3*q^3*r where p, q, and r are prime:
- A179690 Products of 4 distinct primes (p^2*q^2*r*s): (2,2,1,1)
- A179691 Products of 3 distinct primes (p^5*q^2*r): (5,2,1)
- A179693 Products of 4 distinct primes (p^4*q*r*s): (4,1,1,1)
- A179696 Numbers with prime signature {7,1,1}: (7,1,1)
- A179698 Numbers of the form p^4*q^3*r: (4,3,1)
- A179700 Products of 4 distinct primes (p^3*q^2*r*s): (3,2,1,1)
- A179703 Numbers of the form p^6*q^2*r: (6,2,1)
- A179704 Products of 4 distinct primes (p^5*q*r*s): (5,1,1,1)
- A179983 Positive integers n such that, if k appears in n's prime signature, k-1 appears at least as often as k (for any integer k > 1):
- A180925 Natural numbers with palindromic number of divisors:
- A182120 Numbers n for which the canonical prime factorization contains only exponents which are congruent to 2 modulo 3:
- A182358 Numbers n for which the number of divisors of n is congruent to 2 mod 4:
- A182853 Squarefree composite integers and powers of squarefree composite integers:
- A182854 Integers whose prime signature a) contains at least two distinct numbers, and b) contains no number that occurs less often than any other number:
- A182855 Numbers that require exactly five iterations to reach a fixed point under the x -> A181819(x) map:
- A186285 Numbers of the form p^(3^k) where p is prime and k >= 0:
- A187039 Numbers that have equal counts of even and odd exponents of primes in their factorization:
- A188654 Numbers such that in canonical prime factorization the maximal exponent does not equal the number of positive exponents:
- A189975 Numbers with prime factorization pqr^3: (3,1,1)
- A189982 Numbers with prime factorization pqrs^2: (2,1,1,1)
- A189983 Numbers with prime factorization pqrst^2: (2,1,1,1,1)
- A189984 Numbers with prime factorization pqrst^3: (3,1,1,1,1)
- A189987 Numbers with prime factorization pq^6: (6,1)
- A190107 Numbers with prime factorization pqr^2s^4: (4,2,1,1)
- A190641 Numbers having exactly one non-unitary prime factor:
- A190892 Numbers n that can be written as n = a*b = c*d*e, where a, b, c, d, and e are distinct composite numbers:
- A192690 Nonprime numbers with a nonprime number of nonprime divisors:
- A195086 Numbers n such that (number of prime factors of n counted with multiplicity) less (number of distinct prime factors of n) = 2:
- A195087 Numbers n such that (number of prime factors of n counted with multiplicity) less (number of distinct prime factors of n) = 3:
- A195088 Numbers n such that (number of prime factors of n counted with multiplicity) less (number of distinct prime factors of n) = 4:
- A195089 Numbers n such that (number of prime factors of n counted with multiplicity) less (number of distinct prime factors of n) = 5:
- A195090 Numbers n such that (number of prime factors of n counted with multiplicity) less (number of distinct prime factors of n) = 6:
- A195091 Numbers n such that (number of prime factors of n counted with multiplicity) less (number of distinct prime factors of n) = 7:
- A195092 Numbers n such that (number of prime factors of n counted with multiplicity) less (number of distinct prime factors of n) = 8:
- A195093 Numbers n such that (number of prime factors of n counted with multiplicity) less (number of distinct prime factors of n) = 9:
- A197300 The Riemann primes of the theta type and index 4:
- A197680 Numbers whose prime factors have powers that are squares:
- A200511 Numbers n with omega(n)=2 and bigomega(n)>2, where omega=A001221=number of distinct prime factors, bigomega=A001222=prime factors counted with multiplicity:
- A200521 Numbers n such that omega(n)=4 but bigomega(n)>4, i.e., having exactly 4 distinct prime factors, but at least one of these with multiplicity > 1:
- A209061 Exponentially squarefree numbers: (), (1), (2), (1,1), (3), (2,1), (1,1,1), (3,1), (2,2), (2,1,1), (1,1,1,1), (5), (3,2), (3,1,1), (2,2,1), (2,1,1,1), ...
- A210490 Union of squares (A000290) and squarefree numbers (A005117):
- A210994 Numbers n such that A000005(n) <> 4:
- A211337 Numbers n for which the number of divisors, tau(n), is congruent to 1 modulo 3:
- A211338 Numbers n for which the number of divisors, tau(n), is congruent to 2 modulo 3:
- A211484 Numbers n for which the canonical prime factorization contains only an even number of exponents, all of which are congruent to 1 modulo 3:
- A211485 Numbers n for which the canonical prime factorization contains only an odd number of exponents, all of which are congruent to 1 modulo 3:
- A212164 Numbers n such that the maximal exponent in its prime factorization is greater than the number of positive exponents (A051903(n) > A001221(n)):
- A212165 Numbers n such that the maximal exponent in its prime factorization is not less than the number of positive exponents (A051903(n) >= A001221(n)):
- A212166 Numbers n such that the maximal exponent in its prime factorization equals the number of positive exponents (A051903(n) = A001221(n)):
- A212167 Numbers n such that the maximal exponent in its prime factorization is not greater than the number of positive exponents (A051903(n) <= A001221(n)):
- A212168 Numbers n such that the maximal exponent in its prime factorization is less than the number of positive exponents (A051903(n) < A001221(n)):
- A213367 Numbers that are not squares of primes:
- A214195 Numbers with the number of distinct prime factors a multiple of 3:
- A216417 Numbers of the form p^2*q^3 where p, q are (not necessarily distinct) primes:
- A216426 Numbers of the form a^2*b^3, where a != b and a, b > 1:
- A216427 Numbers of the form a^2*b^3, where a >= 2 and b >= 2:
- A216883 Primes p such that x^31 = 2 has a solution mod p:
- A216884 Primes p such that x^61 = 2 has a solution mod p:
- A216885 Primes p such that x^47 = 2 has a solution mod p:
- A216886 Primes p such that x^59 = 2 has a solution mod p:
- A217856 Numbers with three prime factors, not necessarily distinct, except cubes of primes:
- A220218 Numbers where all exponents in its prime factorization are one less than a prime:
- A223456 Composite numbers whose number of proper divisors has a prime number of proper divisors:
- A225228 Numbers with prime signatures (1,1,1) or (2,2,1) or (3,2,2):
- A228056 Numbers of the form p * m^2, where p is prime and m > 1:
- A229125 Numbers of the form p * m^2, where p is prime and m > 0: union of A228056 and A000040:
- A229153 Numbers of the form c * m^2, where m > 0 and c is composite and squarefree:
- A229972 Nonprime numbers n such that the product of their proper divisors is a perfect cube:
- A230843 Cube-free numbers which in their canonical prime factorization have mutually distinct exponents:
- A233182 Numbers that are not the product of a prime and a square:
- A238748 Numbers n such that each integer that appears in the prime signature of n appears an even number of times:
- A239289 Numbers that are not the product of three (not necessarily distinct) primes:
- A245080 Numbers such that omega(a(n)) is a proper divisor of bigomega(a(n)):
- A245303 Product of a prime and a power (exponent at least 2, base at least 1):
- A246547 Prime powers p^e where p is a prime and e >= 2 (prime powers without the primes or 1):
- A246551 Prime powers p^e where p is a prime and e is odd:
- A246655 Numbers of the form p^k where p is a prime and k >= 1:
- A246716 Positive numbers that are not the product of (exactly) two distinct primes:
- A252849 Numbers with an even number of square divisors:
- A252895 Numbers with an odd number of square divisors:
- A253388 Numbers n such that the number of divisors of n is the product of two distinct primes:
- A255429 Numbers n which have a proper number of divisors which is prime:
- A258456 Product of divisors of n is not a square:
- A259183 Complement of A259444:
- A259444 a(1)=2. For n>1, a(n) = smallest number > a(n-1) such that, for all m,r<n, a(n) != a(m)^a(r):

Charles R Greathouse IV, *Sequences determined by prime signature*.— From the On-Line Encyclopedia of Integer Sequences® Wiki (OEIS® Wiki). [https://oeis.org/wiki/Sequences_determined_by_prime_signature]