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

1,1

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.

Links

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

Example

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

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 *)

Crossrefs

Cf. A000142, A000290.

Sequence in context: A083458 A124621 A325237 * A065799 A162582 A123098

Adjacent sequences: A358068 A358069 A358070 * A358072 A358073 A358074

Keyword

nonn

Author

Walter Robinson, Oct 30 2022

Status

approved