A063951 Every number is the sum of 4 squares; these are the odd numbers n such that the first square can be taken to be any positive square < n.

1, 3, 5, 7, 9, 13, 15, 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

Odd numbers n such that for all k with 1 <= k < sqrt(n), n - k^2 is not in A004215. - Robert Israel, Jan 24 2018

The only numbers for which allowing k = 0 would make a difference are 7 and 15: These two are not in A063954.

References

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

Crossrefs

Cf. A004215, A063949, A063950, A063952, A063953, A063954.

Programs

Maple

isA004215:= proc(n)
  local t;
  t:= padic:-ordp(n,2);
  t::even and (n/2^t) mod 8 = 7
end proc:
filter:= proc(n) andmap(not(isA004215), [seq(n - k^2, k=1..floor(sqrt(n-1)))]) end proc:
select(filter, [seq(i,i=1..1000,2)]); # Robert Israel, Jan 24 2018

Mathematica

ok[n_] := Range[ Floor[ Sqrt[n] ]] == DeleteCases[ Union[ Flatten[ PowersRepresentations[n, 4, 2]]], 0, 1, 1]; A063951 = Select[ Range[1, 999, 2], ok] (* Jean-François Alcover, Sep 12 2012 *)

PARI

is_A063951(n)=bittest(n,0)&&!forstep(k=sqrtint(n-1),1,-1,isA004215(n-k^2)&&return) \\ M. F. Hasler, Jan 26 2018

PARI

A063951=select(is_A063951,[1..945]) \\ M. F. Hasler, Jan 26 2018

Formula

This A063951 = A063954 U { 7, 15 }. - M. F. Hasler, Jan 27 2018