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Numbers that have an equal number of factorizations of even and odd length in both unordered and ordered manners.
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%I #82 Jan 02 2022 16:09:45

%S 4,9,12,18,20,25,28,44,45,48,49,50,52,63,68,72,75,76,80,92,98,99,108,

%T 112,116,117,121,124,147,148,153,162,164,169,171,172,175,176,180,188,

%U 192,200,207,208,212,236,240,242,244,245,252,261,268,272,275,279,284,289,292,300

%N Numbers that have an equal number of factorizations of even and odd length in both unordered and ordered manners.

%C Intersection of A319240 and A013929, i.e., terms of A319240 that are not squarefree.

%C A319240 lists the numbers that have an equal number of factorizations of even and odd length in an unordered manner.

%C A013929 lists the numbers that have an equal number of factorizations of even and odd length in an ordered manner.

%C There are infinitely many terms in this sequence since p^2 is always such a number for prime p.

%C 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.

%C 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.

%C Similar methods can be applied to all prime signatures.

%C Wilf's conjecture is equivalent to the statement that this sequence is the set difference of A319240 and A006881.

%D Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005.

%H Valerio De Angelis and Dominic Marcello, <a href="https://www.jstor.org/stable/10.4169/amer.math.monthly.123.6.557">Wilf's Conjecture</a>, The American Mathematical Monthly 123.6 (2016): 557-573.

%H S. de Wannemacker, T. Laffey and R. Osburn, <a href="https://arxiv.org/abs/math/0608085">On a conjecture of Wilf</a>, arXiv:math/0608085 [math.NT], 2006-2007.

%e 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.

%o (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

%o isok(m) = (f(m) == 0) && ! issquarefree(m); \\ _Michel Marcus_, Dec 09 2021

%Y Cf. A006881, A013929, A074206, A319240, A316441, A000587.

%K nonn,easy

%O 1,1

%A _Tian Vlasic_, Dec 05 2021