A349079 Numbers k such that there exists m, 1 <= m <= k with the property that A000178(k) / m! is a square, where A000178(k) = k$ = 1!*2!*...*k! is the superfactorial of k.

1, 2, 4, 8, 12, 14, 16, 18, 20, 24, 28, 32, 34, 36, 40, 44, 48, 52, 56, 60, 62, 64, 68, 72, 76, 80, 84, 88, 92, 96, 98, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 142, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 194, 196, 200, 204, 208, 212, 216, 220, 224, 228

Offset

1,2

Comments

If k is a term, then A348692(k) lists integers m such that k$ / m! is a square; and for each k, there exist only one (A349080) or two (A349081) such integers m.

See A348692 for further information, links and references about Olympiads.

Except for 1, all terms are even, and, when k is such an even term, corresponding m belong(s) to {k/2 - 2, k/2 - 1, k/2, k/2 + 1, k/2 + 2}.

This sequence is the union of {1} and of three infinite and disjoint subsequences:

-> A008586, so every positive multiple of 4 is a term and in this case, for k=4*q, (k$)/(k/2)! = ( 2^(k/4) * Product_{j=1..k/2} ((2j-1)!) )^2 (see example 4).

-> A060626, so every k = 4*q^2 - 2 (q >= 1) is a term (see examples 2 and 14).

-> 2*A055792 = {k = 2q^2 with q>1 in A001541} = {18, 578, ...} (see example 18).

Links

Table of n, a(n) for n=1..66.

Rick Mabry and Laura McCormick, Square products of punctured sequences of factorials, Gaz. Aust. Math. Soc., 2009, pages 346-352.

Index to sequences related to Olympiads and other Mathematical competitions.

Example

2 is a term as 2$ / 2! = 1^2.
4 is a term as 4$ / 2! = 12^2.
14 is a term as 14$ / 8! = 1309248519599593818685440000000^2 and also 14$ / 9! = 436416173199864606228480000000^2.
18 is a term as 18$ / 7! = 29230177671473293820176594405114531928195727360000000000000^2.

Mathematica

supfact[n_] := supfact[n] = BarnesG[n + 2]; fact[n_] := fact[n] = n!; q[k_] := AnyTrue[Range[k], IntegerQ @ Sqrt[supfact[k]/fact[#]] &]; Select[Range[230], q] (* Amiram Eldar, Nov 08 2021 *)

Prog

(PARI) f(n) = prod(k=2, n, k!); \\ A000178
isok(k) = my(sf=f(k)); for (m=1, k, if (issquare(sf/m!), return(1))); \\ Michel Marcus, Nov 08 2021

Crossrefs

Cf. A001541, A000178, A055792, A348692, A349080, A349081.

Subsequences: A008586, A060626.

Sequence in context: A256940 A361833 A048314 * A356053 A118286 A211823

Adjacent sequences: A349076 A349077 A349078 * A349080 A349081 A349082

Keyword

nonn

Author

Bernard Schott, Nov 07 2021

Status

approved