A307715 Decimal expansion of Sum_{t>0} log((t + 1)/t)^2.

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

Offset

0,1

Comments

This constant appears at several places in the literature:

1) In the asymptotic formula of the number of minimal covering systems with exactly n elements (see Theorem 1.1 in Balister, Bollobás, Morris, Sahasrabudhe and Tiba) and

2) in the maximal size of the iterated divisor function

(see Theorem 1 in Buttkewitz, Elsholtz, Ford and Schlage-Puchta) and

3) in the maximal order of the iterated r_2 function, which counts the number of representations as sums of 2 squares (see Theorems 2.1. and 2.3 in Elsholtz, M. Technau and N. Technau). - Modified by C. Elsholtz, Apr 15 2025

Links

Table of n, a(n) for n=0..104.

P. Balister, B. Bollobás, R. Morris, J. Sahasrabudhe and M. Tiba, The structure and number of Erdős covering systems, arXiv:1904.04806 [math.CO], 2019. J. Eur. Math. Soc. 26, 75-109 (2024).

Y. Buttkewitz, C. Elsholtz, K. Ford, and J.-C. Schlage-Puchta, The maximal order of iterated multiplicative functions, arxiv:1108.1815 [math.NT], 2011. IMRN issue 17, 4051-4061 (2012).

C. Elsholtz, M. Technau, and N. Technau, The maximal order of iterated multiplicative functions, arxiv:1709.04799 [math.NT], 2017. Mathematika, 65 (4) 2019, 990-1009.

P. Erdős, Egy kongruenciarendszerekrol szóló problémáról, (On a problem concerning congruence-systems, in Hungarian), Mat. Lapok, 4 (1952), 122-128.

Formula

From Amiram Eldar, Jun 17 2023: (Start)

Equals 2 * Sum_{k>=1} H(k) * (zeta(k+1)-1) / (k+1), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

Equals -Sum_{k>=1} zeta'(2*k) / k. (End)

Example

0.9771891832689365544578857494764347480773925064747239017702...

Mathematica

First[RealDigits[NSum[(Log[(t + 1)/t])^2, {t, 1, Infinity}, NSumTerms -> 100, Method -> {"NIntegrate", "MaxRecursion" -> 10}, WorkingPrecision -> 100]]]

Prog

(PARI) sumpos(t=1, log((t + 1)/t)^2) \\ Michel Marcus, Apr 26 2019

Crossrefs

Cf. A001008, A002805, A010553, A080340, A094076, A275489, A294593, A296195, A335831.

Sequence in context: A381157 A199503 A242714 * A269547 A333345 A183699

Adjacent sequences: A307712 A307713 A307714 * A307716 A307717 A307718

Keyword

nonn,cons

Author

Stefano Spezia, Apr 24 2019

Status

approved