A025358 Numbers that are the sum of 4 nonzero squares in exactly 2 ways.

31, 34, 36, 37, 39, 43, 45, 47, 49, 50, 54, 57, 61, 68, 69, 71, 74, 77, 81, 83, 86, 94, 107, 113, 116, 131, 136, 144, 149, 200, 216, 272, 296, 344, 376, 464, 544, 576, 800, 864, 1088, 1184, 1376, 1504, 1856, 2176, 2304, 3200, 3456, 4352, 4736, 5504, 6016, 7424

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

Conjecture: the even members of this sequence are all numbers of the form

k*4^m for k in [9,17,29], m>= 1, or k*4^m for k in [34, 50, 54, 74, 86, 94], m>=0. - Robert Israel, Nov 03 2017

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..77 (first 71 terms from Robert Price)
• Eric Weisstein's World of Mathematics, Square Number
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A025367 (at least 2 ways).

Programs

Maple

N:= 10000: # to get all terms <= N
T:= Vector(N):
for a from 1 to floor(sqrt(N/4)) do
    for b from a to floor(sqrt((N-a^2)/3)) do
      for c from b to floor(sqrt((N-a^2-b^2)/2)) do
        for d from c to floor(sqrt(N-a^2-b^2-c^2)) do
          m:= a^2+b^2+c^2+d^2;
          T[m]:= T[m]+1;
od od od od:
select(i -> T[i] = 2, [$1..N]); # Robert Israel, Nov 03 2017

Mathematica

M = 1000;
Clear[T]; T[_] = 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;
        T[m] = T[m] + 1;
]]]];
Select[Range[M], T[#] == 2&] (* Jean-François Alcover, Mar 22 2019, after Robert Israel *)

Formula

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