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

Author

V. Raman, Sep 07 2012

Keywords

• nonn
• look

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

Crossrefs

Cf. A001481, A154777, A092572.

Programs

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

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 */