A063950 -id:A063950 - cp's OEIS frontend

A063949 Every number is the sum of 4 squares; these are the numbers n for which the first square can be taken to be any positive square < n.

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, 14, 15, 17, 18, 20, 21, 22, 25, 26, 28, 30, 33, 34, 36, 38, 41, 42, 45, 46, 49, 50, 52, 54, 57, 58, 60, 62, 65, 66, 68, 70, 73, 74, 78, 81, 82, 84, 86, 89, 90, 94, 97, 98, 100, 102, 105, 106, 110, 114, 118, 122, 126, 129, 130

Offset: 1

N. J. A. Sloane, Sep 04 2001

The only primes of this form are 2, 3, 5, 7, 13, 17, 41, 73, 89, 97, 257, 313, 353, 433.

Also, the numbers n such that for no 0 < k < sqrt(n), n-k^2 is in A004215, i.e., of the form 4^i(8j+7). The largest odd number in this sequence is a(322) = 945, cf. A063951. - M. F. Hasler, Jan 26 2018

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

T. D. Noe, Table of n, a(n) for n = 1..1109 (numbers < 4000)

Cf. A063950, A063951, A063952, A063953, A063954.

t1 = {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}; Union[{0}, t1, 4*t1, 4*Range[0, 999] + 2] (* T. D. Noe, Feb 22 2012 *)

is_A063949(n)=if(bittest(n,0),is_A063951(n),n%4==2||is_A063951(n/4)||!n) \\ M. F. Hasler, Jan 26 2018

#A063949_vec=select( is_A063949, [0..3780]) /* or: setunion(setunion(concat(0,A063951), 4*A063951),apply(t->t-2,4*[1..945])) */

A063949(n)=if(n>1054,n*4-438,A063949_vec[n]) \\ M. F. Hasler, Jan 26 2018

Consists of 0, the 54 odd numbers in A063951, 4 times those numbers and all numbers of the form 4m+2.

a(n) = 4*(n-110) + 2 for all n > 1054. - M. F. Hasler, Jan 26 2018

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.

Original entry on oeis.org

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

N. J. A. Sloane, Sep 04 2001

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

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

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

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

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 *)

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

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.

Original entry on oeis.org

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

N. J. A. Sloane, Sep 04 2001

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.

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

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

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

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 *)

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

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

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

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A063949 Every number is the sum of 4 squares; these are the numbers n for which the first square can be taken to be any positive square < n.

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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.

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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.

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