OFFSET
1,1
COMMENTS
First differs from A006881 at a(53) = 180.
By length, we mean the number of factors in a particular factorization.
There are infinitely many terms in this sequence since all squarefree semiprimes (listed in A006881) are always such numbers.
There are no terms of the form p^k with p prime (listed in A000961).
Out of all numbers of the form p*q^k, p and q prime, only the numbers of the form p*q (A006881) and p*q^6 (A189987) are terms.
Similar methods can be applied to all prime signatures.
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
6=2*3 (unrestricted) has an equal number (1) of even-length factorizations and odd-length factorizations, and 6=2*3 (distinct) has an equal number (1) of even-length factorizations and odd-length factorizations.
MATHEMATICA
facs[n_] := If[n <= 1, {{}}, Join @@ Table[Map[Prepend[#, d] &, Select[facs[n/d], Min @@ # >= d &]], {d, Rest[Divisors[n]]}]]; Intersection @@ First@Flatten[Position[#, 0] & /@ Transpose@Table[Sum[(-1)^Length[f], {f, #}] & /@ {facs[n], Select[facs[n], UnsameQ @@ # &]}, {n, #1, #2}], {3}]&[1, 194] (* Robert P. P. McKone, Jan 05 2022, from Gus Wiseman in A319238 and A319240 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Tian Vlasic, Jan 01 2022
STATUS
approved