A331593 Numbers k that have the same number of distinct prime factors as A225546(k).

1, 2, 3, 4, 5, 7, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 28, 29, 31, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 101, 103, 104, 107, 108, 109, 112, 113, 116, 117, 121, 124, 127, 131, 135, 136, 137, 139, 144, 147, 148, 149

Offset

1,2

Comments

Numbers k for which A001221(k) = A331591(k).

Numbers k that have the same number of terms in their factorization into powers of distinct primes as in their factorization into powers of squarefree numbers with distinct exponents that are powers of 2. See A329332 for a description of the relationship between the two factorizations and A225546.

If k is included, then all such x that A046523(x) = k are also included, i.e., all numbers with the same prime signature as k. Notably, primes (A000040) are included, but squarefree semiprimes (A006881) are not.

k^2 is included if and only if k is included, for example A001248 is included, but A085986 is not.

Links

Table of n, a(n) for n=1..78.

Formula

{a(n)} = {k : A001221(k) = A000120(A267116(k))}.

Example

There are 2 terms in the factorization of 36 into powers of distinct primes, which is 36 = 2^2 * 3^2 = 4 * 9; but only 1 term in its factorization into powers of squarefree numbers with distinct exponents that are powers of 2, which is 36 = 6^(2^1). So 36 is not included.
There are 2 terms in the factorization of 40 into powers of distinct primes, which is 40 = 2^3 * 5^1 = 8 * 5; and also 2 terms in its factorization into powers of squarefree numbers with distinct exponents that are powers of 2, which is 40 = 10^(2^0) * 2^(2^1) = 10 * 4. So 40 is included.

Mathematica

Select[Range@ 150, Equal @@ PrimeNu@ {#, If[# == 1, 1, Apply[Times, Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]]]} &] (* Michael De Vlieger, Jan 26 2020 *)

Prog

(PARI)
A331591(n) = if(1==n, 0, my(f=factor(n), u=#binary(vecmax(f[, 2])), xs=vector(u), m=1, e); for(i=1, u, for(k=1, #f~, if(bitand(f[k, 2], m), xs[i]++)); m<<=1); #select(x -> (x>0), xs));
k=0; n=0; while(k<105, n++; if(omega(n)==A331591(n), k++; print1(n, ", ")));

Crossrefs

Cf. A000120, A046523, A225546, A267116, A329332.

Sequences with related definitions: A001221, A331591, A331592.

Subsequences: A000040, A001248, A050376, A054753, A065036, A162142, A225547.

Subsequences of complement: A006881, A056824, A085986, A120944, A177492.

Sequence in context: A117290 A286972 A210994 * A319237 A331230 A258456

Adjacent sequences: A331590 A331591 A331592 * A331594 A331595 A331596

Keyword

nonn

Author

Antti Karttunen and Peter Munn, Jan 21 2020

Status

approved