A353337 - OEIS

A353337

Number of ways to write n as a product of the terms of A028260 larger than 1; a(1) = 1 by convention (an empty product).

1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 1, 1, 0, 2, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 3, 0, 1, 1, 2, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 2, 1, 2, 1, 1, 0, 3, 0, 1, 0, 3, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 1, 0, 3, 1, 1, 1, 2, 0, 3, 1, 0, 1, 1, 1, 4, 0, 0, 0, 3, 0, 0, 0, 2, 0

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OFFSET

1,16

COMMENTS

Number of factorizations of n into factors k > 1 for which there is an even number of primes (when counted with multiplicity, A001222) in their prime factorization.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from exponents in factorization of n

FORMULA

a(n) = a(A046523(n)). [The sequence depends only on the prime signature of n].

For all n >= 1, a(n) >= A320655(n), and a(n) >= A353377(n).

EXAMPLE

Of the eleven divisors of 96 larger than one, the following: [4, 6, 16, 24, 96] are terms of A028260 because they have an even number of prime factors when counted with repetition. Using them, we can factor 96 in four possible ways, as 96 = 24*4 = 16*6 = 6*4*4, therefore a(96) = 4.

PROG

(PARI)

A065043(n) = (1 - (bigomega(n)%2));

A353337(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&A065043(d), s += A353337(n/d, d))); (s));

CROSSREFS

Cf. A001222, A028260, A038548, A046523, A065043, A353338 [= a(n^2)].

Cf. also A320655, A353377.

Sequence in context: A334944 A174875 A193510 * A351907 A357374 A264048

Adjacent sequences: A353334 A353335 A353336 * A353338 A353339 A353340

KEYWORD

nonn

AUTHOR

Antti Karttunen, Apr 17 2022

STATUS

approved