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 A362984 Decimal expansion of the asymptotic mean of the abundancy index of the powerful numbers (A001694). 2
 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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS The abundancy index of a positive integer k is A000203(k)/k = A017665(k)/A017666(k). 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). LINKS Table of n, a(n) for n=1..105. Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (6). FORMULA Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A180114(k)/A001694(k). 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). EXAMPLE 2.14968690301526765128219042105109416145987653275100999873... MATHEMATICA \$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]] PROG (PARI) prodeulerrat((p^8 + p^4 + p^3 - 1)/(p^8 - p^2), 1/2) CROSSREFS Cf. A000203, A001694, A017665, A017666, A180114. 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). Sequence in context: A163236 A325008 A325016 * A077160 A228043 A262066 Adjacent sequences: A362981 A362982 A362983 * A362985 A362986 A362987 KEYWORD nonn,cons AUTHOR Amiram Eldar, May 12 2023 STATUS approved

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Last modified September 7 16:30 EDT 2024. Contains 375749 sequences. (Running on oeis4.)