A245058 -id:A245058 - cp's OEIS frontend

A362985 Decimal expansion of the asymptotic mean of the abundancy index of the cubefull numbers (A036966).

2, 4, 8, 2, 1, 7, 9, 1, 9, 6, 4, 2, 2, 3, 5, 9, 5, 2, 5, 4, 6, 1, 6, 7, 6, 4, 3, 6, 7, 4, 6, 8, 7, 6, 9, 8, 5, 3, 6, 3, 6, 8, 9, 4, 0, 9, 7, 1, 9, 3, 0, 4, 6, 8, 3, 5, 4, 3, 6, 3, 9, 3, 2, 8, 1, 4, 4, 4, 2, 3, 3, 8, 8, 5, 7, 6, 7, 5, 0, 4, 6, 3, 4, 1, 1, 5, 0, 7, 3, 1, 0, 3, 9, 8, 0, 4, 4, 7, 4, 0, 3, 7, 3, 1, 0

Offset: 1

Amiram Eldar, May 12 2023

			2.48217919642235952546167643674687698536368940971930468354...

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).

Cf. A000203, A036966, A017665, A017666, A362986.

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), A245058 (even), A240976 (squares), A306633 (squarefree), A362984 (powerful).

$MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{2, -1, -2, 3, -2, -1, 3, -2, -2, 3, -1, -2, 3, -1, -1, 1}, {0, 0, 0, -4, 0, 6, 7, 4, 9, 0, -11, -22, -26, -21, -15, 20}, m]; RealDigits[((2^5 + 2^(10/3) + 2^3 + 2^(8/3) - 1)/(2^(10/3)*(2^(5/3) + 2^(1/3) + 1)))*((3^5 + 3^(10/3) + 3^3 + 3^(8/3) - 1)/(3^(10/3)*(3^(5/3) + 3^(1/3) + 1))) * Zeta[4/3] * Exp[NSum[Indexed[c, n]*(PrimeZetaP[n/3] - 1/2^(n/3) - 1/3^(n/3))/n, {n, 4, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 120][[1]]

zeta(4/3) * prodeulerrat((p^15 + p^10 + p^9 + p^8 - 1)/(p^10 * (p^5 + p + 1)), 1/3)

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A362986(k)/A036966(k).

Equals zeta(4/3) * Product_{p prime} ((p^5 + p^(10/3) + p^3 + p^(8/3) - 1)/(p^(10/3) * (p^(5/3) + p^(1/3) + 1))).

A137933 Least common multiple of n^2 and 2.

Original entry on oeis.org

2, 4, 18, 16, 50, 36, 98, 64, 162, 100, 242, 144, 338, 196, 450, 256, 578, 324, 722, 400, 882, 484, 1058, 576, 1250, 676, 1458, 784, 1682, 900, 1922, 1024, 2178, 1156, 2450, 1296, 2738, 1444, 3042, 1600, 3362, 1764, 3698, 1936, 4050, 2116, 4418, 2304, 4802, 2500, 5202, 2704

Offset: 1

William A. Tedeschi, Feb 29 2008

nonn

Cf. A016742, A077591, A109043, A245058.

LCM[Range[60]^2,2] (* Harvey P. Dale, Jun 06 2018 *)

a(n) = lcm(n^2, 2); \\ Michel Marcus, Mar 13 2018

a(n) = lcm(n^2, 2).
From R. J. Mathar, Mar 06 2008: (Start)
O.g.f.: -2x(1 + 6x^2 + x^4 + 2x^3 + 2x)/((-1+x)^3 * (x+1)^3).
a(2n) = A016742(n).
a(2n+1) = A077591(n). (End)
a(n) = n*A109043(n). - Michel Marcus, Mar 13 2018
From Amiram Eldar, Jul 06 2022: (Start)
Sum_{n>=1} 1/a(n) = 5*Pi^2/48.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/48 = -A245058. (End)

cp's OEIS Frontend

A362985 Decimal expansion of the asymptotic mean of the abundancy index of the cubefull numbers (A036966).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula