A350486 Numbers that have an equal number of even- and odd-length unordered factorizations and also an equal number of even- and odd-length unordered factorizations into distinct factors.

6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 180, 183, 185, 187, 192, 194

Offset: 1

Author

Tian Vlasic, Jan 01 2022

Keywords

• nonn
• easy

Comments

First differs from A006881 at a(53) = 180.

By length, we mean the number of factors in a particular factorization.

Intersection of A319240 (factors are not necessarily distinct) and A319238 (factors are distinct).

Numbers k such that A316441(k) = A114592(k) = 0.

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.

Examples

			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.
		

Links

• Leonhard Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
• Leonhard Euler, De mirabilibus proprietatibus numerorum pentagonalium, par. 2
• Eric Weisstein's World of Mathematics, Pentagonal Number Theorem
• Wikipedia, Pentagonal number theorem

Crossrefs

Cf. A006881, A114592, A189987, A316441, A319240, A319238.

Programs

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 *)