A379812 a(n) = sigma_1(n) * sigma_2(n).
1, 15, 40, 147, 156, 600, 400, 1275, 1183, 2340, 1464, 5880, 2380, 6000, 6240, 10571, 5220, 17745, 7240, 22932, 16000, 21960, 12720, 51000, 20181, 35700, 32800, 58800, 25260, 93600, 30784, 85995, 58560, 78300, 62400, 173901, 52060, 108600, 95200, 198900, 70644
Offset: 1
References
- Srinivasa Ramanujan, Collected papers, edited by G. H. Hardy et al., Chelsea, 1962, chapter 17, pp. 133-135.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
- R. Balasubramanian and K. Ramachandra, The place of an identity of Ramanujan in prime number theory, Proceedings of the Indian Academy of Sciences, Section A, Vol. 83. No. 4 (1976), pp. 156-165.
- Yoichi Motohashi, On an identity of Ramanujan, Hardy-Ramanujan Journal, Vol. 41 (2019), pp. 48-49.
- Srinivasa Ramanujan, Some formulae in the analytic theory of numbers, Messenger of Mathematics, Vol. 45 (1916), pp. 81-84.
- Bertram Martin Wilson, Proofs of some formulae enunciated by Ramanujan, Proceedings of the London Mathematical Society, Vol. s2-21, No. 1 (1923), pp. 235-255.
Crossrefs
Programs
-
Mathematica
a[n_] := Times @@ DivisorSigma[{1, 2}, n]; Array[a, 50] -
PARI
a(n) = {my(f = factor(n)); sigma(f) * sigma(f, 2);}
Formula
Multiplicative with a(p^e) = (p^(e+1)-1) * (p^(2*e+2)-1) / ((p-1) * (p^2-1)).
Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(s-2) * zeta(s-3) / zeta(2*s-3).
In general, Dirichlet g.f. of sigma_i(n) * sigma_j(n): zeta(s) * zeta(s-i) * zeta(s-j) * zeta(s-i-j) / zeta(2*s-i-j) (Ramanujan, 1916).
Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(2) * zeta(3) * zeta(4) / zeta(5) = zeta(3) * Pi^6 / (540*zeta(5)) = 2.06386841111121962734... .
In general, Sum_{k=1..n} sigma_i(k) * sigma_j(k) ~ c(i,j) * n^(i+j+1) / (i+j+1), for i, j >= 1, where c(i,j) = zeta(i+1) * zeta(j+1) * zeta(i+j+1) / zeta(i+j+2).
G.f.: Sum_{k>=1} Sum_{l>=1} k*l^2*x^lcm(k, l)/(1 - x^lcm(k, l)). - Miles Wilson, Jul 10 2025