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

Author

N. J. A. Sloane, Sep 04 2001

Keywords

• nonn
• easy
• nice

Comments

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

References

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

Links

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

Crossrefs

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

Programs

Mathematica

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 *)

PARI

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

PARI

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

PARI

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

Formula

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