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
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
KEYWORD
nonn
AUTHOR
Bernard Schott, Dec 01 2021
STATUS
approved