A308041 - OEIS

A308041

Decimal expansion of lim_{m->oo} (1/log(m))*Sum_{k=1..m} 1/usigma(k), where usigma(k) is the sum of unitary divisors of k (A034448).

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

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OFFSET

0,1

LINKS

Table of n, a(n) for n=0..105.

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 51 (constant Y3).

V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocals of some multiplicative functions - II, Indian J. Pure Appl. Math., Vol. 11 (1980), pp. 1334-1355 (eq. 3.8-3.9, p. 1352-1353).

László Tóth, Alternating sums concerning multiplicative arithmetic functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1 (section 4.9, p. 29).

FORMULA

From Amiram Eldar, Dec 23 2024: (Start)

Equals Product_{p prime} ((1-1/p) * (1 + Sum_{k>=1} 1/(p^k+1))).

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A063974(k). (End)

EXAMPLE

0.76871836244648519867273433245535052523425574041190...

MATHEMATICA

$MaxExtraPrecision = 1000; m = 1000; f[p_] := 1 - (p - 1)/p*Sum[1/p^k/(p^k + 1), {k, 1, m}]; c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]*Range[0, m]]; RealDigits[f[2]*Exp[NSum[Indexed[c, k]*(PrimeZetaP[k] - 1/2^k)/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

CROSSREFS

Cf. A034448, A063974, A308039 (corresponding limit with sigma).

Sequence in context: A196397 A238301 A154170 * A256685 A372445 A019325

Adjacent sequences: A308038 A308039 A308040 * A308042 A308043 A308044

KEYWORD

nonn,cons

AUTHOR

Amiram Eldar, May 10 2019

EXTENSIONS

More digits from Vaclav Kotesovec, Jun 13 2021

STATUS

approved