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 A362984 #7 May 12 2023 04:22:34
%S A362984 2,1,4,9,6,8,6,9,0,3,0,1,5,2,6,7,6,5,1,2,8,2,1,9,0,4,2,1,0,5,1,0,9,4,
%T A362984 1,6,1,4,5,9,8,7,6,5,3,2,7,5,1,0,0,9,9,9,8,7,3,2,7,3,3,4,3,7,8,9,7,6,
%U A362984 2,7,1,7,9,4,0,3,6,4,2,3,6,5,7,4,2,7,4,2,3,7,7,1,7,0,2,4,2,2,8,9,7,3,8,6,2
%N A362984 Decimal expansion of the asymptotic mean of the abundancy index of the powerful numbers (A001694).
%C A362984 The abundancy index of a positive integer k is A000203(k)/k = A017665(k)/A017666(k).
%C A362984 The asymptotic mean of the abundancy index over all the positive integers is lim_{m->oo} (1/m) * Sum_{k=1..m} A000203(k)/k = Pi^2/6 = zeta(2) = 1.644934... (A013661).
%H A362984 Rafael Jakimczuk and Matilde Lalín, <a href="https://doi.org/10.7546/nntdm.2022.28.4.617-634">Asymptotics of sums of divisor functions over sequences with restricted factorization structure</a>, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (6).
%F A362984 Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A180114(k)/A001694(k).
%F A362984 Equals Product_{p prime} (p^4 + p^2 + p^(3/2) - 1)/(p^4 - p) = Product_{p prime} (1 + (p^2 + p^(3/2) + p - 1)/(p^4 - p)) (Jakimczuk and Lalín, 2022).
%e A362984 2.14968690301526765128219042105109416145987653275100999873...
%t A362984 $MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{2, -3, 4, -6, 7, -7, 7, -6, 5, -3, 2, -1}, {0, 0, 0, 4, 5, 6, 0, -12, -9, -5, 0, 22}, m]; RealDigits[(2^4 + 2^2 + 2^(3/2) - 1)/(2^4 - 2)*(3^4 + 3^2 + 3^(3/2) - 1)/(3^4 - 3) * Exp[NSum[Indexed[c, n]*(PrimeZetaP[n/2] - 1/2^(n/2) - 1/3^(n/2))/n, {n, 4, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 120][[1]]
%o A362984 (PARI) prodeulerrat((p^8 + p^4 + p^3 - 1)/(p^8 - p^2), 1/2)
%Y A362984 Cf. A000203, A001694, A017665, A017666, A180114.
%Y A362984 Similar constants (the asymptotic mean of the abundancy index of other sequences): A013661 (all positive integers), A082020 (cubefree), A111003 (odd), A157292 (5-free), A157294 (7-free), A157296 (9-free), A240976 (squares), A245058 (even), A306633 (squarefree), A362985 (cubefull).
%K A362984 nonn,cons
%O A362984 1,1
%A A362984 _Amiram Eldar_, May 12 2023