A349931 Numbers that have an equal number of factorizations of even and odd length in both unordered and ordered manners.

4, 9, 12, 18, 20, 25, 28, 44, 45, 48, 49, 50, 52, 63, 68, 72, 75, 76, 80, 92, 98, 99, 108, 112, 116, 117, 121, 124, 147, 148, 153, 162, 164, 169, 171, 172, 175, 176, 180, 188, 192, 200, 207, 208, 212, 236, 240, 242, 244, 245, 252, 261, 268, 272, 275, 279, 284, 289, 292, 300

Offset: 1

Author

Tian Vlasic, Dec 05 2021

Keywords

• nonn
• easy

Comments

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

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.

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

Examples

			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.
		

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.

Crossrefs

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

Programs

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