A063954 Every number is the sum of 4 squares; these are the odd numbers such that the first square can be taken to be any square < n.

1, 3, 5, 9, 13, 17, 21, 25, 33, 41, 45, 49, 57, 65, 73, 81, 89, 97, 105, 129, 145, 153, 169, 177, 185, 201, 209, 217, 225, 257, 273, 297, 305, 313, 329, 345, 353, 385, 425, 433, 441, 481, 513, 561, 585, 609, 689, 697, 713, 817, 825, 945

Offset: 1

Author

N. J. A. Sloane, Sep 04 2001

Keywords

• nonn
• easy
• nice
• fini
• full

Comments

This sequence is finite, with 52 terms.

This is a variant of A063951 where the arbitrary first squares must be positive. This makes a difference only for n = 7 and n = 15, which are in A063951 but not in this sequence, because for these two n and k = 0, n - k^2 is in A004215, i.e., not the sum of fewer than 4 squares. - M. F. Hasler, Jan 27 2018

References

• J. H. Conway, personal communication, Aug 27, 2001.

Links

• Gordon Pall, On Sums of Two or Four Values of a Quadratic Function of x, Transactions of the American Mathematical Society, Vol. 34, No. 1, (January 1932), pp. 98-125. - _Ant King_, Nov 01 2010

Crossrefs

Cf. A063949, A063950, A063951, A063952, A063953.

Programs

Mathematica

j[k_] := If[Union[Flatten[PowersRepresentations[k,4,2]]^2] == (#^2&/@Range[0,Sqrt[k]]), True, False]; Select[Range[1,1250,2], j] (* Ant King, Nov 01 2010 *)

PARI

is_A063954(n)=bittest(n, 0)&&!forstep(k=sqrtint(n-1), 0, -1, isA004215(n-k^2)&&return) \\ M. F. Hasler, Jan 27 2018