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%I #28 Dec 11 2022 00:47:35
%S 2,6,10,124,145,220,649,745,1081,1249,1345,2929,3601,3745,5065,5076,
%T 5161,5209,5481,6049,6196,6265,6804,7249,7945,8289,9529,11124,14644,
%U 15649,17361,17809,21169,22921,30649,35316,40321,40384,40720,40761,43456,43569,43801
%N Numbers k that can be written as the sum of a perfect square and a factorial in at least 2 distinct ways.
%C This does not count x^2 and (-x)^2 as distinct, nor does it count 0! and 1! as distinct.
%C 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.
%e 145 = 5^2 + 5! = 11^2 + 4! = 12^2 + 1!.
%t With[{f = Range[8]!}, c[n_] := Count[f, _?(IntegerQ @ Sqrt[n - #] &)]; Select[Range[f[[-1]]], c[#] > 1 &]] (* _Amiram Eldar_, Oct 30 2022 *)
%Y Cf. A000142, A000290.
%K nonn
%O 1,1
%A _Walter Robinson_, Oct 30 2022