A395407 Decimal expansion of Sum_{k>=1} H(k)/(k+1)!, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

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

Offset

0,1

Links

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

J. Hammond, Problem 6609, Mathematical Questions with Their Solutions: From the "Educational Times", Vol. 60 (1894), p. viii; Solution by H. J. Woodall, ibid., pp. 81-82.

Leroy Quet, Problem 10398, The American Mathematical Monthly, Vol. 101, No. 7 (1994), p. 682; The Effect of an Alternating Series, Solution to Problem 10398 by Michael Vowe, ibid., Vol. 104, No. 5 (1997), p. 462.

Michael I. Shamos, A catalog of the real numbers, 2007, p. 602.

Formula

Equals e * Sum_{k>=1} H(k)*(-1)^(k+1)/(k+1)! (Quet, 1994).

Equals gamma - Ei(1) + Sum_{k>=1} H(k) / k! = A001620 - A091725 + A347952.

Equals Integral_{x=0..1} exp(x)*log(x/(1-x)) dx (Hammond, 1894).

Formulas from Shamos (2007):

Equals gamma * (1 + e) - Ei(1) - e*Ei(-1) = A001620 * (1 + A001113) - A091725 - A001113*(-A099285).

Equals Sum_{k>=2} (gamma - psi(k))/k!, where psi is the digamma function.

Equals Sum_{k>=1} Sun_{m=1..k} 1/(m*(k+1)!) = e * Sum_{k>=1} Sun_{m=1..k} (-1)^(k+1)/(m*(k+1)!).

Example

0.847480063872532464560977504243198730406519521948659...

Mathematica

RealDigits[EulerGamma * (1 + E) - ExpIntegralEi[1] - E * ExpIntegralEi[-1], 10, 120][[1]]

Prog

(PARI) Euler * (1 + exp(1)) - real(-eint1(-1)) + exp(1) * eint1(1)

Crossrefs

Cf. A001113, A001008, A001620, A002805, A073003, A091725, A099285, A229837, A347952.

Sequence in context: A242023 A249415 A021122 * A388924 A388239 A389048

Adjacent sequences: A395404 A395405 A395406 * A395408 A395409 A395410

Keyword

nonn,cons

Author

Amiram Eldar, Apr 21 2026

Status

approved