A369118 k is a term if and only if k is a composite number where the bases and the exponents of its factors in the prime decomposition are all odd primes.

27, 125, 243, 343, 1331, 2187, 2197, 3125, 3375, 4913, 6859, 9261, 12167, 16807, 24389, 29791, 30375, 35937, 42875, 50653, 59319, 68921, 78125, 79507, 83349, 84375, 103823, 132651, 148877, 161051, 166375, 177147, 185193, 205379, 226981, 273375, 274625

Offset

1,1

Comments

Every term is divisible by a cube.

If n and k are terms then n*k is a term if and only if gcd(n, k) = 1.

From Michael De Vlieger, Jan 19 2024: (Start)

Prime exponents of prime power factors p^m | k imply that k is a powerful number. Hence this sequence is a proper subset of A001694, and k is of the form a^2 * b^3.

Prime exponents m imply either perfect powers k in A001597 such that all the m are the same, or an Achilles number k (in A052486) if the exponents differ. This is because prime p divides itself but is coprime to primes q != p. Therefore this sequence is not a subsequence of A001597.

The sequence consists of composite prime powers (A246547) and powerful numbers that are not prime powers (A286708), both of which are numbers that are not squarefree (A013929). (End)

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Formula

Sum_{n>=1} 1/a(n) = -1 + Product_{prime >= 3} (1 + Sum_{prime q >= 3} 1/p^q) = 0.05534030537711484966... . - Amiram Eldar, Mar 08 2025

Example

25015118625 = 3^5 * 5^3 * 7^7 is a term.
3125 = 5^5 and 3375 = 3^3 * 5^3 are terms but 3125*3375 is not a term.

Mathematica

A369118Q[n_] := OddQ[n] && AllTrue[FactorInteger[n], OddQ[#] && PrimeQ[#]&, 2];
Select[Range[500000], A369118Q] (* Paolo Xausa, Jan 19 2024 *)

Prog

(SageMath)
def isA369118(n):
    return (n > 1 and is_odd(n) and all(is_odd(f[1]) and is_prime(f[1])
           for f in factor(n)))
print([n for n in range(1, 300000) if isA369118(n)])
(PARI) isok(k) = k > 1 && (k % 2 && #select(x -> (x <= 2) || !isprime(x), factor(k)[, 2]) == 0); \\ Amiram Eldar, Mar 08 2025

Crossrefs

Cf. A002808 (superset), A001694 (superset).

A051674 is a subsequence for n>1.

Intersects A001597, A013929, A052486, A246547, A286708.

Subsequence of A381825.

Sequence in context: A125497 A179145 A381825 * A118092 A371189 A126272

Adjacent sequences: A369115 A369116 A369117 * A369119 A369120 A369121

Keyword

nonn

Author

Peter Luschny, Jan 17 2024

Status

approved