OFFSET
1,1
COMMENTS
From Tian Vlasic, Jan 01 2022: (Start)
Numbers that have an equal number of even- and odd-length unordered factorizations into distinct factors.
For prime p, by the pentagonal number theorem, p^k is a term if and only if k is in A090864.
For primes p and q, p*q^k is a term if and only if k = A000326(m)+N with 0 <= N < m. (End)
LINKS
L. Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
L. Euler, De mirabilibus proprietatibus numerorum pentagonalium, par. 2
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem
Wikipedia, Pentagonal number theorem
EXAMPLE
16 = 2*8 = 4*4 = 2*2*4 = 2*2*2*2 has an equal number of even-length factorizations and odd-length factorizations into distinct factors (1). - Tian Vlasic, Dec 31 2021
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Join@@Position[Table[Sum[(-1)^Length[f], {f, Select[facs[n], UnsameQ@@#&]}], {n, 100}], 0]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 15 2018
STATUS
approved