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%I #43 Nov 22 2021 12:26:40
%S 1,2,4,8,12,14,16,18,20,24,28,32,34,36,40,44,48,52,56,60,62,64,68,72,
%T 76,80,84,88,92,96,98,100,104,108,112,116,120,124,128,132,136,140,142,
%U 144,148,152,156,160,164,168,172,176,180,184,188,192,194,196,200,204,208,212,216,220,224,228
%N 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.
%C If k is 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.
%C See A348692 for further information, links and references about Olympiads.
%C 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}.
%C This sequence is the union of {1} and of three infinite and disjoint subsequences:
%C -> A008586, so every positive multiple of 4 is a term and in this case, for k=4*q, (k$)/(k/2)! = ( 2^(k/4) * Product_{j=1..k/2} ((2j-1)!) )^2 (see example 4).
%C -> A060626, so every k = 4*q^2 - 2 (q >= 1) is a term (see examples 2 and 14).
%C -> 2*A055792 = {k = 2q^2 with q>1 in A001541} = {18, 578, ...} (see example 18).
%H Rick Mabry and Laura McCormick, <a href="https://www.austms.org.au/wp-content/uploads/Gazette/2009/Nov09/TechPaperMabry.pdf">Square products of punctured sequences of factorials</a>, Gaz. Aust. Math. Soc., 2009, pages 346-352.
%H <a href="/index/O#Olympiads">Index to sequences related to Olympiads and other Mathematical competitions</a>.
%e 2 is a term as 2$ / 2! = 1^2.
%e 4 is a term as 4$ / 2! = 12^2.
%e 14 is a term as 14$ / 8! = 1309248519599593818685440000000^2 and also 14$ / 9! = 436416173199864606228480000000^2.
%e 18 is a term as 18$ / 7! = 29230177671473293820176594405114531928195727360000000000000^2.
%t 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 *)
%o (PARI) f(n) = prod(k=2, n, k!); \\ A000178
%o isok(k) = my(sf=f(k)); for (m=1, k, if (issquare(sf/m!), return(1))); \\ _Michel Marcus_, Nov 08 2021
%Y Cf. A001541, A000178, A055792, A348692, A349080, A349081.
%Y Subsequences: A008586, A060626.
%K nonn
%O 1,2
%A _Bernard Schott_, Nov 07 2021
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