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A358071
Numbers k that can be written as the sum of a perfect square and a factorial in at least 2 distinct ways.
1
2, 6, 10, 124, 145, 220, 649, 745, 1081, 1249, 1345, 2929, 3601, 3745, 5065, 5076, 5161, 5209, 5481, 6049, 6196, 6265, 6804, 7249, 7945, 8289, 9529, 11124, 14644, 15649, 17361, 17809, 21169, 22921, 30649, 35316, 40321, 40384, 40720, 40761, 43456, 43569, 43801
OFFSET
1,1
COMMENTS
This does not count x^2 and (-x)^2 as distinct, nor does it count 0! and 1! as distinct.
For any two factorials a > b, where a-b = m*n where m > n and (m and n are both even or m and n are both odd), (((m-n)/2)^2 + a) will appear in this sequence.
EXAMPLE
145 = 5^2 + 5! = 11^2 + 4! = 12^2 + 1!.
MATHEMATICA
With[{f = Range[8]!}, c[n_] := Count[f, _?(IntegerQ @ Sqrt[n - #] &)]; Select[Range[f[[-1]]], c[#] > 1 &]] (* Amiram Eldar, Oct 30 2022 *)
CROSSREFS
Sequence in context: A083458 A124621 A325237 * A065799 A162582 A123098
KEYWORD
nonn
AUTHOR
Walter Robinson, Oct 30 2022
STATUS
approved