%I #17 Dec 04 2021 12:17:10

%S 8,14,16,32,48,72,96,128,160,200,240,288,336,392,448,512,574,576,648,

%T 720,800,880,968,1056,1152,1248,1352,1456,1568,1680,1800,1920,2048,

%U 2176,2312,2448,2592,2736,2888,3040,3200,3360,3528,3696,3872,4048,4232,4416,4608,4800,5000

%N Numbers k for which there exist two integers m with 1 <= m_1 < m_2 <= k such that A000178(k) / m! is a square, where A000178(k) = k$ = 1!*2!*...*k! is the superfactorial of k.

%C This sequence is the union of three infinite and disjoint subsequences:

%C -> Numbers k = 8t^2 > 0 (A139098); for these numbers, m_1 = k/2 - 1 = 4t^2-1 < m_2 = k/2 = 4t^2 (see example for k = 8).

%C -> Numbers k = 8t*(t+1) (A035008); for these numbers, m_1 = k/2 = 4t(t+1) < m_2 = k/2 + 1 = (2t+1)^2 (see example for k = 16).

%C -> Even numbers of the form 2t^2-4, t>1 in A001541 (A349766); for these numbers, m_1 = k/2 + 1 = t^2 - 1 < m_2 = k/2 + 2 = t^2 (see example for k = 14).

%C See A348692 for further information.

%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.

%e For k = 8, 8$ / 2! is not a square, but m_1 = 3 because 8$ / 3! = 29030400^2 and m_2 = 4 because 8$ / 4! = 14515200^2.

%e For k = 14, m_1 = 8 because 14$ / 8! = 1309248519599593818685440000000^2 and m_2 = 9 because 14$ / 9! = 436416173199864606228480000000^2.

%e For k = 16, m_1 = 8 because 16$ / 8! = 6848282921689337839624757371207680000000000^2 and m_2 = 9 because 16$ / 9! = 2282760973896445946541585790402560000000000^2.

%t Do[j=0;l=1;g=BarnesG[k+2];While[j<2&&l<=k,If[IntegerQ@Sqrt[g/l!],j++];l++];If[j==2,Print@k],{k,5000}] (* _Giorgos Kalogeropoulos_, Dec 02 2021 *)

%o (PARI) sf(n) = prod(k=2, n, k!); \\ A000178

%o isok(m) = if (!(m%2), my(s=sf(m)); #select(issquare, vector(4, k, s/(m/2+k-2)!), 1) == 2); \\ _Michel Marcus_, Dec 04 2021

%Y Cf. A001541, A035008, A139098, A348692, A349080.

%Y Subsequence of A349079.

%K nonn

%O 1,1

%A _Bernard Schott_, Dec 01 2021