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.

1, 2, 4, 12, 18, 20, 24, 28, 34, 36, 40, 44, 52, 56, 60, 62, 64, 68, 76, 80, 84, 88, 92, 98, 100, 104, 108, 112, 116, 120, 124, 132, 136, 140, 142, 144, 148, 152, 156, 164, 168, 172, 176, 180, 184, 188, 192, 194, 196, 204, 208, 212, 216, 220, 224, 228, 232, 236, 244, 248, 252, 254, 256

Offset

1,2

Comments

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

-> 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).

-> 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)

-> 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).

See A348692 for further information.

Links

Table of n, a(n) for n=1..63.

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

Example

For k = 2, 2$ / 2! = 1^2, hence 2 is a term.
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.
For k = 18 and m = 7, we have 18$ / 7! = 29230177671473293820176594405114531928195727360000000000000^2 and there is no other solution m, hence 18 is a term.

Mathematica

q[n_] := Count[BarnesG[n + 2]/Range[n]!, _?(IntegerQ@Sqrt[#] &)] == 1; Select[Range[100], q] (* Amiram Eldar, Nov 20 2021 *)

Prog

(PARI) sf(n) = prod(k=2, n, k!); \\ A000178
isok(m) = my(s=sf(m)); #select(issquare, vector(m, k, s/k!), 1) == 1; \\ Michel Marcus, Nov 20 2021

Crossrefs

Cf. A000178, A001541, A060626, A348692, A182834, A349496.

Subsequence of A349079.

Sequence in context: A085931 A125885 A067929 * A337837 A064361 A303403

Adjacent sequences: A349077 A349078 A349079 * A349081 A349082 A349083

Keyword

nonn

Author

Bernard Schott, Nov 20 2021

Status

approved