A275489 -id:A275489

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A307715

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

+10
0

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

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OFFSET

0,1

COMMENTS

This constant appears 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).

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.

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, A080340, A094076, A275489, A294593, A296195.

KEYWORD

nonn,cons

AUTHOR

Stefano Spezia, Apr 24 2019

STATUS

approved

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