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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.

%I A063951 #27 Jan 30 2018 21:10:47
%S A063951 1,3,5,7,9,13,15,17,21,25,33,41,45,49,57,65,73,81,89,97,105,129,145,
%T A063951 153,169,177,185,201,209,217,225,257,273,297,305,313,329,345,353,385,
%U A063951 425,433,441,481,513,561,585,609,689,697,713,817,825,945
%N 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.
%C A063951 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
%C A063951 The only numbers for which allowing k = 0 would make a difference are 7 and 15: These two are not in A063954.
%D A063951 J. H. Conway, personal communication, Aug 27, 2001.
%F A063951 This A063951 = A063954 U { 7, 15 }. - _M. F. Hasler_, Jan 27 2018
%p A063951 isA004215:= proc(n)
%p A063951   local t;
%p A063951   t:= padic:-ordp(n,2);
%p A063951   t::even and (n/2^t) mod 8 = 7
%p A063951 end proc:
%p A063951 filter:= proc(n) andmap(not(isA004215), [seq(n - k^2, k=1..floor(sqrt(n-1)))]) end proc:
%p A063951 select(filter, [seq(i,i=1..1000,2)]); # _Robert Israel_, Jan 24 2018
%t A063951 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 *)
%o A063951 (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
%o A063951 (PARI) A063951=select(is_A063951,[1..945]) \\ _M. F. Hasler_, Jan 26 2018
%Y A063951 Cf. A004215, A063949, A063950, A063952, A063953, A063954.
%K A063951 nonn,easy,nice,fini,full
%O A063951 1,2
%A A063951 _N. J. A. Sloane_, Sep 04 2001