A025367 Numbers that are the sum of 4 nonzero squares in 2 or more ways.

28, 31, 34, 36, 37, 39, 42, 43, 45, 47, 49, 50, 52, 54, 55, 57, 58, 60, 61, 63, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 102, 103, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A000414, A024796, A025358, A025368, A025406, A344795.

Programs

Maple

N:= 1000: # to get all terms <= N
V:= Vector(N):
for x from 1 while x^2 +3 <= N do
for y from 1 to x while x^2 + y^2 + 2 <= N do
  for z from 1 to y while x^2 + y^2 + z^2 + 1 <= N do
    for w from 1 to z while x^2 + y^2 + z^2 + w^2 <= N do
       t:= x^2 + y^2 + z^2 + w^2;
       V[t]:= V[t]+1;
od od od od:
select(t -> V[t] >= 2, [$1..N]); # Robert Israel, Jul 05 2017

Mathematica

Select[Range@ 200, 2 == Length@ Quiet@ IntegerPartitions[#, {4}, Range[Sqrt@ #]^2, 2] &] (* Giovanni Resta, Jul 05 2017 *)
M = 1000;
Clear[V]; V[_] = 0;
For[a = 1, a <= Floor[Sqrt[M/4]], a++,
  For[b = a, b <= Floor[Sqrt[(M - a^2)/3]], b++,
    For[c = b, c <= Floor[Sqrt[(M - a^2 - b^2)/2]], c++,
      For[d = c, d <= Floor[Sqrt[M - a^2 - b^2 - c^2]], d++,
        m = a^2 + b^2 + c^2 + d^2;
        V[m] = V[m] + 1;
]]]];
Select[Range[M], V[#] >= 2&] (* Jean-François Alcover, Mar 22 2019, after Robert Israel *)

Formula

{n: A025428(n) >= 2}. - R. J. Mathar, Jun 15 2018