This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A279889 #36 Jan 07 2025 01:59:01
%S A279889 0,1,66,1577,18218,135550,738236,3207785,11714718,37347144,106499470,
%T A279889 277489886,668981686,1512360404,3228797252,6570019945,12793050456,
%U A279889 24001960051,43483452090,76485144056,130752372320,218220937122,355664809556,568293832670,889969136158
%N A279889 a(n) = Sum_{k=1..n-1} sigma_5(k)*sigma_5(n-k).
%C A279889 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.
%D A279889 Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory,Springer-Verlag, 1976. See p. 140, exercise 10.
%D A279889 Srinivasa Ramanujan, Collected papers, ed. G. H. Hardy et al., Cambridge, 1927, pp. 136-162.
%H A279889 Seiichi Manyama, <a href="/A279889/b279889.txt">Table of n, a(n) for n = 1..1000</a>
%H A279889 William Duke, <a href="https://www.math.ucla.edu/~wdduke/preprints/ramanujan.pdf">Ramanujan and modular forms</a>, in: K. Alladi et al., Srinivasa Ramanujan: His Life, Legacy, and Mathematical Influence, Springer Cham, 2024. See eq. (6).
%H A279889 D. H. Lehmer, Some functions of Ramanujan, Math. Student, Vol. 27 (1959), pp. 105-116; <a href="https://schoolbooksarchive.azimpremjiuniversity.edu.in/handle/20.500.12497/11781">entire volume</a>. See p. 111, eq. (9).
%H A279889 Srinivasa Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram18.html">On certain arithmetical functions</a>, Trans. Cambridge Philos. Soc., Vol. 22, No. 9 (1916), pp. 159-184.
%F A279889 A027860(n) = (sigma_11(n) - sigma_5(n) + 252*a(n))/756.
%t A279889 a[n_] := (65 * DivisorSigma[11, n] + 691 * DivisorSigma[5, n] - 756 * RamanujanTau[n]) / 174132; Array[a, 25] (* _Amiram Eldar_, Jan 07 2025 *)
%o A279889 (PARI) a(n) = sum(k=1, n-1, sigma(k, 5)*sigma(n-k, 5)) \\ _Felix Fröhlich_, Jan 01 2017
%o A279889 (PARI) a(n) = {my(f = factor(n)); (65 * sigma(f, 11) + 691 * sigma(f, 5) - 756 * ramanujantau(n)) / 174132;} \\ _Amiram Eldar_, Jan 07 2025
%Y A279889 Cf. Sum_{k=1..n-1} sigma_m(k)*sigma_m(n-k): A087115 (m=3), this sequence (m=5).
%Y A279889 Cf. A000594, A001160, A013959, A027860.
%K A279889 nonn
%O A279889 1,3
%A A279889 _Seiichi Manyama_, Dec 22 2016