A349081 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.

8, 14, 16, 32, 48, 72, 96, 128, 160, 200, 240, 288, 336, 392, 448, 512, 574, 576, 648, 720, 800, 880, 968, 1056, 1152, 1248, 1352, 1456, 1568, 1680, 1800, 1920, 2048, 2176, 2312, 2448, 2592, 2736, 2888, 3040, 3200, 3360, 3528, 3696, 3872, 4048, 4232, 4416, 4608, 4800, 5000

Offset

1,1

Comments

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

-> 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).

-> 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).

-> 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).

See A348692 for further information.

Links

Table of n, a(n) for n=1..51.

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

Example

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.
For k = 14, m_1 = 8 because 14$ / 8! = 1309248519599593818685440000000^2 and m_2 = 9 because 14$ / 9! = 436416173199864606228480000000^2.
For k = 16, m_1 = 8 because 16$ / 8! = 6848282921689337839624757371207680000000000^2 and m_2 = 9 because 16$ / 9! = 2282760973896445946541585790402560000000000^2.

Mathematica

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 *)

Prog

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

Crossrefs

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

Subsequence of A349079.

Sequence in context: A235143 A330705 A179186 * A192045 A084021 A138666

Adjacent sequences: A349078 A349079 A349080 * A349082 A349083 A349084

Keyword

nonn

Author

Bernard Schott, Dec 01 2021

Status

approved