A329266 -id:A329266 - cp's OEIS frontend

A329264 a(n) is the number of solutions of the infinite Diophantine equation Sum_{j>0} j^r*(k_j)^2 = n with k_j integers and r = 2.

1, 2, 0, 0, 4, 4, 0, 0, 4, 4, 4, 0, 0, 12, 8, 0, 6, 16, 4, 0, 16, 8, 8, 0, 8, 24, 20, 0, 0, 52, 24, 0, 12, 32, 28, 8, 24, 12, 48, 16, 24, 68, 48, 8, 16, 96, 32, 16, 8, 68, 96, 32, 40, 68, 128, 32, 80, 88, 76, 48, 32, 156, 104, 64, 8, 224, 192, 40, 88, 152, 208

Offset: 0

Stefano Spezia, Nov 09 2019

nonn

			a(16) = 6 since there are 6 integer solutions to 1^2*k1^2 + 2^2*k2^2 + 3^2*k3^2 + 4^2*k4^2 + ... = 16:
k1 = +-4 and k_j = 0 for j > 1;
k1 = 0, k2 = +-2 and k_j = 0 for j > 2;
k1 = k2 = k3 = 0, k4 = +-1 and k_j = 0 for j > 4.

Nian Hong Zhou, Yalin Sun, Counting the number of solutions to certain infinite Diophantine equations, arXiv:1910.07884 [math.NT], 2019.

Cf. A000041, A000122, A320067 (r = 1), A320068, A320078, A320968, A320992, A329265 (r = 3), A329266 (r = 4).

nmax=70; r=2; CoefficientList[Series[Product[Product[(1-(-1)^n*q^(n*j^r))/(1+(-1)^n*q^(n*j^r)),{n,1,nmax}],{j,1,nmax}],{q,0,nmax}],q]

a(n) = [q^n] Product_{j>0} Product_{n>0} (1 - (-1)^n*q^(n*j^r)) / (1 + (-1)^n*q^(n*j^r)) with r = 2 (see Proposition 1.1 in Zhou and Sun).

A329265 a(n) is the number of solutions of the infinite Diophantine equation Sum_{j>0} j^r*(k_j)^2 = n with k_j integers and r = 3.

Original entry on oeis.org

1, 2, 0, 0, 2, 0, 0, 0, 2, 6, 0, 0, 4, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 4, 2, 0, 2, 4, 0, 0, 4, 2, 8, 0, 4, 18, 0, 0, 8, 0, 4, 0, 4, 12, 0, 0, 0, 4, 2, 0, 8, 4, 0, 0, 0, 0, 8, 0, 4, 16, 0, 0, 12, 4, 4, 0, 0, 16, 0, 0, 8, 10, 16, 0, 8, 16, 0, 0, 0, 4, 18, 0, 0, 16

Offset: 0

Stefano Spezia, Nov 09 2019

nonn

			a(9) = 6 since there are 6 integer solutions to 1^3*k1^2 + 2^3*k2^2 + ... = 9:
k1 = +-3 and k_j = 0 for j > 1;
k1 = -1, k2 = +-1 and k_j = 0 for j > 2;
k1 = 1, k2 = +-1 and k_j = 0 for j > 2.

Nian Hong Zhou, Yalin Sun, Counting the number of solutions to certain infinite Diophantine equations, arXiv:1910.07884 [math.NT], 2019.

Cf. A000041, A000122, A320067 (r = 1), A320068, A320078, A320968, A320992, A329264 (r = 2), A329266 (r = 4).

nmax=85; r=3; CoefficientList[Series[Product[Product[(1-(-1)^n*q^(n*j^r))/(1+(-1)^n*q^(n*j^r)),{n,1,nmax}],{j,1,nmax}],{q,0,nmax}],q]

a(n) = [q^n] Product_{j>0} Product_{n>0} (1 - (-1)^n*q^(n*j^r)) / (1 + (-1)^n*q^(n*j^r)) with r = 3 (see Proposition 1.1 in Zhou and Sun).

cp's OEIS Frontend

A329264 a(n) is the number of solutions of the infinite Diophantine equation Sum_{j>0} j^r*(k_j)^2 = n with k_j integers and r = 2.

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