A395407 - OEIS

%I #7 Apr 21 2026 09:07:39

%S 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,

%T 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,

%U 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

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

%H J. Hammond, <a href="https://archive.org/details/mathematicalque00unkngoog/page/n148/mode/1up">Problem 6609</a>, Mathematical Questions with Their Solutions: From the "Educational Times", Vol. 60 (1894), p. viii; <a href="https://archive.org/details/mathematicalque00unkngoog/page/n221/mode/1up">Solution</a> by H. J. Woodall, ibid., pp. 81-82.

%H Leroy Quet, <a href="https://doi.org/10.2307/2974700">Problem 10398</a>, The American Mathematical Monthly, Vol. 101, No. 7 (1994), p. 682; <a href="https://doi.org/10.2307/2974747">The Effect of an Alternating Series</a>, Solution to Problem 10398 by Michael Vowe, ibid., Vol. 104, No. 5 (1997), p. 462.

%H Michael I. Shamos, <a href="https://euro.ecom.cmu.edu/people/faculty/mshamos/cat.pdf">A catalog of the real numbers</a>, 2007, p. 602.

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

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

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

%F Formulas from Shamos (2007):

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

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

%F 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)!).

%e 0.847480063872532464560977504243198730406519521948659...

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

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

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

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, Apr 21 2026