A358071 Numbers k that can be written as the sum of a perfect square and a factorial in at least 2 distinct ways.

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

Author

Walter Robinson, Oct 30 2022

Keywords

• nonn

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.

Examples

			145 = 5^2 + 5! = 11^2 + 4! = 12^2 + 1!.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..3661

Crossrefs

Cf. A000142, A000290.

Programs

Maple

Nf:= 15: # for terms <= (Nf) + 1!
Facts:= [seq(i!, i=1..Nf)]:
N:= Facts[-1]+1:
R:= 'R':
for i from 0 to isqrt(N) do
  for j from 1 to Nf do
    v:= Facts[j]+i^2;
    if v > N then break fi;
    if not assigned(R[v]) then R[v]:= [i,j]
    else R[v]:= R[v],[i,j]
    fi
od od:
J:= [indices(R)]:
sort(map(op, select(t -> nops([R[op(t)]])>=2, J))); # Robert Israel, Aug 08 2025

Mathematica

With[{f = Range[8]!}, c[n_] := Count[f, ?(IntegerQ @ Sqrt[n - #] &)]; Select[Range[f[[-1]]], c[#] > 1 &]] (* _Amiram Eldar, Oct 30 2022 *)