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

1,1

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.

Links

Table of n, a(n) for n=1..58.

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

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

Sequence in context: A265693 A211484 A339561 * A346014 A006881 A030229

Adjacent sequences: A350483 A350484 A350485 * A350487 A350488 A350489

Keyword

nonn,easy

Author

Tian Vlasic, Jan 01 2022

Status

approved