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 A002127 M2770 N1114 #49 Aug 01 2025 04:51:21
%S A002127 1,3,9,15,30,45,67,99,135,175,231,306,354,465,540,681,765,945,1040,
%T A002127 1305,1386,1695,1779,2205,2290,2754,2835,3438,3480,4185,4272,5076,
%U A002127 5004,6100,5985,7155,7154,8325,8190,9840,9471,11241,11055,12870,12420,14911
%N A002127 MacMahon's generalized sum of divisors function.
%C A002127 Number of partitions of n with two designated summands. For example: a(5) = 9 because there are 9 partitions of 5 with two designated summands: [4'+ 1'], [3'+ 2'], [3'+ 1'+ 1], [3'+ 1 + 1'], [2'+ 2 + 1'], [2 + 2'+ 1'], [2'+ 1'+ 1 + 1], [2'+ 1 + 1'+ 1], [2'+ 1 + 1 + 1']. - _Omar E. Pol_, Jul 23 2025
%D A002127 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002127 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002127 John Cerkan, <a href="/A002127/b002127.txt">Table of n, a(n) for n = 3..10000</a>
%H A002127 G. E. Andrews and S. C. F. Rose, <a href="http://arxiv.org/abs/1010.5769">MacMahon's sum-of-divisors functions, Chebyshev polynomials, and Quasi-modular forms</a>, arXiv:1010.5769 [math.NT], 2010.
%H A002127 William Craig, Jan-Willem van Ittersum and Ken Ono, <a href="https://www.pnas.org/doi/10.1073/pnas.2409417121">Integer partitions detect the primes</a>, PNAS, Vol. 121, No. 39 (2024), e2409417121.
%H A002127 P. A. MacMahon, <a href="http://dx.doi.org/10.1112/plms/s2-19.1.75">Divisors of numbers and their continuations in the theory of partitions</a>, Proc. London Math. Soc., 19 (1921), 75-113; Coll. Papers II, pp. 303-341.
%H A002127 S. Rose, <a href="http://mathoverflow.net/questions/41457/">What literature is known about MacMahon's generalized sum-of-divisors function?</a>
%F A002127 G.f.: (Sum_{k>=0} (-1)^k * (2*k + 1) * binomial( k+2, 4) * x^( k*(k+1) / 2 )) / (5 * Sum_{k>=0} (-1)^k * (2*k + 1) * x^( k*(k+1) / 2 )). - _Michael Somos_, Jan 10 2012
%F A002127 a(n) = (n^2 - 3*n + 2) * A000203(n) / 8 iff n is an odd prime (see Craig link et al.).
%F A002127 Sum_{k=1..n} a(k) ~ Pi^4 * n^4 / (4!*5!). - _Vaclav Kotesovec_, Aug 01 2025
%e A002127 x^3 + 3*x^4 + 9*x^5 + 15*x^6 + 30*x^7 + 45*x^8 + 67*x^9 + 99*x^10 + ...
%t A002127 A002127[n_] := (DivisorSigma[3, n] - (2*n - 1)*DivisorSigma[1, n])/8;
%t A002127 Array[A002127, 50, 3] (* _Paolo Xausa_, Jul 04 2025, after _Michael Somos_'s PARI *)
%o A002127 (PARI) {a(n) = if( n<1, 0, ( sigma( n, 3) - (2*n - 1) * sigma(n) ) / 8)} /* _Michael Somos_, Jan 10 2012 */
%Y A002127 A diagonal of A060043.
%Y A002127 Cf. A002128.
%Y A002127 Column 2 of A385001.
%K A002127 nonn,easy
%O A002127 3,2
%A A002127 _N. J. A. Sloane_
%E A002127 More terms from _Vladeta Jovovic_, Nov 11 2001