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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. 6

%I

%S 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,

%T 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,

%U 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

%N 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.

%C 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.

%C 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.

%C 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.

%C 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.

%C a(n) = 0 if and only if n is a square. - _Charles R Greathouse IV_, Sep 11 2012

%o (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 */

%Y Cf. A001481, A154777, A092572.

%K nonn

%O 1,2

%A _V. Raman_, Sep 07 2012

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Last modified November 19 06:34 EST 2019. Contains 329310 sequences. (Running on oeis4.)