A216504 Number of values of k for which n can be written in the form a^2 + k*b^2, a >= 0, b >= 0, k > 0. a(n) = 0 if a(n) is infinite.

0, 2, 2, 0, 3, 3, 3, 5, 0, 4, 4, 5, 6, 4, 4, 0, 7, 6, 6, 7, 6, 6, 5, 8, 0, 6, 7, 8, 9, 6, 7, 10, 8, 8, 6, 0, 11, 7, 7, 11, 12, 7, 9, 11, 11, 8, 7, 11, 0, 9, 8, 13, 12, 10, 8, 12, 12, 10, 9, 11, 15, 8, 11, 0, 12, 9, 11, 15, 12, 10, 9, 17, 18, 10, 11, 16, 12, 9, 12, 15, 0, 12, 10, 14, 14, 11, 10, 17, 18, 13, 11, 15, 15, 12, 10, 17, 21, 12, 14, 0

Offset

1,2

Comments

A number can be written as a^2+b^2 if and only if it has no prime factor congruent to 3 (mod 4) raised to an odd power.

A number can be written as a^2+2*b^2 if and only if it has no prime factor congruent to 5 (mod 8) or 7 (mod 8) raised to an odd power.

A number can be written as a^2+3*b^2 if and only if it has no prime factor congruent to 2 (mod 3) raised to an odd power.

A number can be written as a^2+7*b^2 if and only if it has no prime factor congruent to 3 (mod 7) or 5 (mod 7) or 6 (mod 7) raised to an odd power. Also the power of 2 should not be 1, if it can be written in the form a^2+7*b^2.

a(n) = 0 if and only if n is a square. - Charles R Greathouse IV, Sep 11 2012

Links

Table of n, a(n) for n=1..100.

Prog

(PARI) for(n=1, 100, sol=0; for(k=1, n, for(x=0, n, if(issquare(n-k*x*x)&&n-k*x*x>=0, sol++; break))); if(issquare(n), print1(0", "), print1(sol", "))) /* V. Raman, Oct 16 2012 */

Crossrefs

Cf. A001481, A154777, A092572.

Sequence in context: A295675 A135356 A259016 * A216673 A363532 A207383

Adjacent sequences: A216501 A216502 A216503 * A216505 A216506 A216507

Keyword

nonn

Author

V. Raman, Sep 07 2012

Status

approved