OFFSET
1,1
COMMENTS
A319240 lists the numbers that have an equal number of factorizations of even and odd length in an unordered manner.
A013929 lists the numbers that have an equal number of factorizations of even and odd length in an ordered manner.
There are infinitely many terms in this sequence since p^2 is always such a number for prime p.
Out of all numbers of the form p^k with p prime (listed in A000961), only the numbers of the form p^2 (A001248) are terms.
Out of all numbers of the form p*q^k, p and q prime, only the numbers of the form p*q (A006881), p*q^2 (A054753), p*q^4 (A178739) and p*q^6 (A189987) are terms.
Similar methods can be applied to all prime signatures.
REFERENCES
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005.
LINKS
Valerio De Angelis and Dominic Marcello, Wilf's Conjecture, The American Mathematical Monthly 123.6 (2016): 557-573.
S. de Wannemacker, T. Laffey and R. Osburn, On a conjecture of Wilf, arXiv:math/0608085 [math.NT], 2006-2007.
EXAMPLE
12 = 2*6 = 3*4 = 2*2*3 (unordered) has an equal number (2) of even-length factorizations and odd-length factorizations, and 12 = 2*6 = 6*2 = 3*4 = 4*3 = 2*2*3 = 2*3*2 = 3*2*2 (ordered) has an equal number (4) of even-length factorizations and odd-length factorizations.
PROG
(PARI) f(n, m=n, k=0) = if(1==n, (-1)^k, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += f(n/d, d, k+1))); (s)); \\ A316441
isok(m) = (f(m) == 0) && ! issquarefree(m); \\ Michel Marcus, Dec 09 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Tian Vlasic, Dec 05 2021
STATUS
approved