A279889 a(n) = Sum_{k=1..n-1} sigma_5(k)*sigma_5(n-k).

0, 1, 66, 1577, 18218, 135550, 738236, 3207785, 11714718, 37347144, 106499470, 277489886, 668981686, 1512360404, 3228797252, 6570019945, 12793050456, 24001960051, 43483452090, 76485144056, 130752372320, 218220937122, 355664809556, 568293832670, 889969136158

Offset

1,3

Comments

In 1916, Ramanujan found the following identity. tau(n) = sigma_11(n) - 691/756 * (sigma_11(n) - sigma_5(n) + 252 * a(n)). This implies tau(n) == sigma_11(n) mod 691.

References

Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory,Springer-Verlag, 1976. See p. 140, exercise 10.

Srinivasa Ramanujan, Collected papers, ed. G. H. Hardy et al., Cambridge, 1927, pp. 136-162.

Links

Seiichi Manyama, Table of n, a(n) for n = 1..1000

William Duke, Ramanujan and modular forms, in: K. Alladi et al., Srinivasa Ramanujan: His Life, Legacy, and Mathematical Influence, Springer Cham, 2024. See eq. (6).

D. H. Lehmer, Some functions of Ramanujan, Math. Student, Vol. 27 (1959), pp. 105-116; entire volume. See p. 111, eq. (9).

Srinivasa Ramanujan, On certain arithmetical functions, Trans. Cambridge Philos. Soc., Vol. 22, No. 9 (1916), pp. 159-184.

Formula

A027860(n) = (sigma_11(n) - sigma_5(n) + 252*a(n))/756.

Mathematica

a[n_] := (65 * DivisorSigma[11, n] + 691 * DivisorSigma[5, n] - 756 * RamanujanTau[n]) / 174132; Array[a, 25] (* Amiram Eldar, Jan 07 2025 *)

Prog

(PARI) a(n) = sum(k=1, n-1, sigma(k, 5)*sigma(n-k, 5)) \\ Felix Fröhlich, Jan 01 2017
(PARI) a(n) = {my(f = factor(n)); (65 * sigma(f, 11) + 691 * sigma(f, 5) - 756 * ramanujantau(n)) / 174132; } \\ Amiram Eldar, Jan 07 2025

Crossrefs

Cf. Sum_{k=1..n-1} sigma_m(k)*sigma_m(n-k): A087115 (m=3), this sequence (m=5).

Cf. A000594, A001160, A013959, A027860.

Sequence in context: A278850 A104673 A251047 * A241799 A269498 A133318

Adjacent sequences: A279886 A279887 A279888 * A279890 A279891 A279892

Keyword

nonn

Author

Seiichi Manyama, Dec 22 2016

Status

approved