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A349080 Numbers k for which there exists only one integer m with 1 <= m <= k such that A000178(k) / m! is a square, where A000178(k) = k$ = 1!*2!*...*k! is the superfactorial of k.

%I A349080 #26 Dec 15 2021 11:42:47
%S A349080 1,2,4,12,18,20,24,28,34,36,40,44,52,56,60,62,64,68,76,80,84,88,92,98,
%T A349080 100,104,108,112,116,120,124,132,136,140,142,144,148,152,156,164,168,
%U A349080 172,176,180,184,188,192,194,196,204,208,212,216,220,224,228,232,236,244,248,252,254,256
%N A349080 Numbers k for which there exists only one integer m with 1 <= m <= k such that A000178(k) / m! is a square, where A000178(k) = k$ = 1!*2!*...*k! is the superfactorial of k.
%C A349080 This sequence is the union of {1} and of three infinite and disjoint subsequences.
%C A349080 -> Numbers k divisible by 4 but not of the form 8q^2 or 8q(q+1) = {4, 12, 20, 24, 28, ...} (see A182834). For these numbers, the corresponding unique m = k/2 (see example for k = 4).
%C A349080 -> Even numbers k not divisible by 4 and of the form k = 2*A055792 = 2*q^2, q>1 in A001541 = {18, 578, ...}. For these numbers, the corresponding unique m = k/2 - 2 = q^2-2 (see example for k = 18)
%C A349080 -> Even numbers k not divisible by 4, that are in A060626 but not of the form k=2q^2-4 with q>1 in A001541 = {2, 34, 62, 98, 142, 194, ...} (A349496). For these numbers, the corresponding unique m = k/2 + 1 (see example for k = 2).
%C A349080 See A348692 for further information.
%H A349080 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 A349080 For k = 2, 2$ / 2! = 1^2, hence 2 is a term.
%e A349080 For k = 4, 4$ /1! = 288, 4$ / 3! = 48, 4$ / 4! = 12 but for m = 2, 4$ / 2! = 12^2, hence 4 is a term.
%e A349080 For k = 18 and m = 7, we have 18$ / 7! = 29230177671473293820176594405114531928195727360000000000000^2 and there is no other solution m, hence 18 is a term.
%t A349080 q[n_] := Count[BarnesG[n + 2]/Range[n]!, _?(IntegerQ@Sqrt[#] &)] == 1; Select[Range[100], q] (* _Amiram Eldar_, Nov 20 2021 *)
%o A349080 (PARI) sf(n) = prod(k=2, n, k!); \\ A000178
%o A349080 isok(m) = my(s=sf(m)); #select(issquare, vector(m, k, s/k!), 1) == 1; \\ _Michel Marcus_, Nov 20 2021
%Y A349080 Cf. A000178, A001541, A060626, A348692, A182834, A349496.
%Y A349080 Subsequence of A349079.
%K A349080 nonn
%O A349080 1,2
%A A349080 _Bernard Schott_, Nov 20 2021