A297493 -id:A297493 - cp's OEIS frontend

A297491 a(n) = (1/2) * Sum_{|k|<=2sqrt(p)} k^4H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime.

9, 44, 234, 664, 2628, 4354, 9774, 13660, 24264, 48690, 59488, 101194, 137718, 158884, 207504, 297594, 410580, 453778, 601324, 715608, 777814, 985840, 1143324, 1409670, 1825054, 2060298, 2185144, 2449764, 2589730, 2885454, 4096384, 4495788, 5142294

Offset: 1

Seiichi Manyama, Dec 31 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000
N. Lygeros, O. Rozier, A new solution to the equation tau(p) == 0 (mod p), J. Int. Seq. 13 (2010) # 10.7.4.

(1/2) * Sum_{|k|<=2*sqrt(p)} k^m*H(4*p-k^2): A000040 (m=0), A084920 (m=2), this sequence (m=4), A297492 (m=6), A297493 (m=8), A297494 (m=10).

Cf. A259825.

Let b(n) = 2*n^3 - 3*n - 1.

a(n) = b(prime(n)).

A297492 a(n) = (1/2) * Sum_{|k|<=2sqrt(p)} k^6H(4*p-k^2) where H() is the Hurwitz class number and p is the n-th prime.

Original entry on oeis.org

33, 308, 2874, 11528, 72060, 141218, 414918, 648260, 1394328, 3528690, 4608800, 9358298, 14113470, 17077148, 24378288, 39426858, 60555180, 69195410, 100714868, 127012680, 141942878, 194693840, 237229188, 313639470, 442561238, 520209690, 562658408, 655294428

Offset: 1

Seiichi Manyama, Dec 31 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000
N. Lygeros, O. Rozier, A new solution to the equation tau(p) == 0 (mod p), J. Int. Seq. 13 (2010) # 10.7.4.

(1/2) * Sum_{|k|<=2*sqrt(p)} k^m*H(4*p-k^2): A000040 (m=0), A084920 (m=2), A297491 (m=4), this sequence (m=6), A297493 (m=8), A297494 (m=10).

Cf. A259825.

b(n) = 5*n^4 - 9*n^2 - 5*n - 1;
a(n) = b(prime(n)); \\ Michel Marcus, Jan 01 2018

Let b(n) = 5*n^4 - 9*n^2 - 5*n - 1.

a(n) = b(prime(n)).

A297494 a(n) = (1/2) * Sum_{|k|<=2sqrt(p)} k^10H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime.

Original entry on oeis.org

513, 20708, 584874, 4714408, 72449100, 200562418, 1012788198, 1953009460, 6172747128, 24788658690, 37242612640, 107770200778, 198936710910, 265200653548, 449592659568, 931777815258, 1775665528380, 2155635964450, 3812897562148, 5368106367720, 6351988507678

Offset: 1

Seiichi Manyama, Dec 31 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000
N. Lygeros, O. Rozier, A new solution to the equation tau(p) == 0 (mod p), J. Int. Seq. 13 (2010) # 10.7.4.
Eric Weisstein's World of Mathematics, Tau Function.

(1/2) * Sum_{|k|<=2*sqrt(p)} k^m*H(4*p-k^2): A000040 (m=0), A084920 (m=2), A297491 (m=4), A297492 (m=6), A297493 (m=8), this sequence (m=10).

Cf. A000594, A259825, A295645, A297127.

Let b(n) = 42*n^6 - 90*n^4 - 75*n^3 - 35*n^2 - 9*n - 1.
a(n) = b(prime(n)) - tau(prime(n)) where tau(n)=A000594(n) is Ramanujan's tau function.
So tau(prime(n)) + 1 == -a(n) (mod prime(n)).

cp's OEIS Frontend

A297491 a(n) = (1/2) * Sum_{|k|<=2sqrt(p)} k^4H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime.

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Formula

A297492 a(n) = (1/2) * Sum_{|k|<=2sqrt(p)} k^6H(4*p-k^2) where H() is the Hurwitz class number and p is the n-th prime.

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Author

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PARI

Formula

A297494 a(n) = (1/2) * Sum_{|k|<=2sqrt(p)} k^10H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime.

Views

Author

Keywords

Links

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Formula

cp's OEIS Frontend

A297491 a(n) = (1/2) * Sum_{|k|<=2*sqrt(p)} k^4*H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime.

Views

Author

Keywords

Links

Crossrefs

Formula

A297492 a(n) = (1/2) * Sum_{|k|<=2*sqrt(p)} k^6*H(4*p-k^2) where H() is the Hurwitz class number and p is the n-th prime.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A297494 a(n) = (1/2) * Sum_{|k|<=2*sqrt(p)} k^10*H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime.

Views

Author

Keywords

Links

Crossrefs

Formula

A297491 a(n) = (1/2) * Sum_{|k|<=2sqrt(p)} k^4H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime.

A297492 a(n) = (1/2) * Sum_{|k|<=2sqrt(p)} k^6H(4*p-k^2) where H() is the Hurwitz class number and p is the n-th prime.

A297494 a(n) = (1/2) * Sum_{|k|<=2sqrt(p)} k^10H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime.