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 there are infinitely many such k.

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

Robert Israel, Table of n, a(n) for n = 1..10000

Maple

N:= 100: # for a(1)..a(N)
m:= floor(sqrt(N)):
V:= Vector(N, i->{}):
for a from 0 to m do
  for b from 1 to m do
    for k from 1 to floor((N-a^2)/b^2) do
      x:= a^2 + k*b^2;
      V[x]:= V[x] union {k};
od od od:
for i from 1 to N do
  if issqr(i) then V[i]:=0 else V[i]:= nops(V[i]) fi
od:
convert(V, list); # Robert Israel, Mar 06 2025

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: A135356 A259016 A379225 * A216673 A363532 A207383

Adjacent sequences: A216501 A216502 A216503 * A216505 A216506 A216507

Keyword

nonn,look

Author

V. Raman, Sep 07 2012

Status

approved