A379812 - OEIS

%I #10 Jan 03 2025 09:36:59

%S 1,15,40,147,156,600,400,1275,1183,2340,1464,5880,2380,6000,6240,

%T 10571,5220,17745,7240,22932,16000,21960,12720,51000,20181,35700,

%U 32800,58800,25260,93600,30784,85995,58560,78300,62400,173901,52060,108600,95200,198900,70644

%N a(n) = sigma_1(n) * sigma_2(n).

%D Srinivasa Ramanujan, Collected papers, edited by G. H. Hardy et al., Chelsea, 1962, chapter 17, pp. 133-135.

%H Amiram Eldar, <a href="/A379812/b379812.txt">Table of n, a(n) for n = 1..10000</a>

%H R. Balasubramanian and K. Ramachandra, <a href="https://www.ias.ac.in/describe/article/seca/083/04/0156-0165">The place of an identity of Ramanujan in prime number theory</a>, Proceedings of the Indian Academy of Sciences, Section A, Vol. 83. No. 4 (1976), pp. 156-165.

%H Yoichi Motohashi, <a href="https://doi.org/10.46298/hrj.2019.5105">On an identity of Ramanujan</a>, Hardy-Ramanujan Journal, Vol. 41 (2019), pp. 48-49.

%H Srinivasa Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram17.html">Some formulae in the analytic theory of numbers</a>, Messenger of Mathematics, Vol. 45 (1916), pp. 81-84.

%H Bertram Martin Wilson, <a href="https://doi.org/10.1112/plms/s2-21.1.235">Proofs of some formulae enunciated by Ramanujan</a>, Proceedings of the London Mathematical Society, Vol. s2-21, No. 1 (1923), pp. 235-255.

%F a(n) = A000203(n) * A001157(n).

%F Multiplicative with a(p^e) = (p^(e+1)-1) * (p^(2*e+2)-1) / ((p-1) * (p^2-1)).

%F Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(s-2) * zeta(s-3) / zeta(2*s-3).

%F 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).

%F 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... .

%F 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).

%t a[n_] := Times @@ DivisorSigma[{1, 2}, n]; Array[a, 50]

%o (PARI) a(n) = {my(f = factor(n)); sigma(f) * sigma(f, 2);}

%Y Cf. A000203, A001157, A072861, A086666, A092346, A158274, A158275, A356533, A356534, A379813, A379814.

%K nonn,easy,mult

%O 1,2

%A _Amiram Eldar_, Jan 03 2025