A262856 Numerators of the Nielsen-Jacobsthal series leading to Euler's constant.

1, 43, 20431, 2150797323119, 9020112358835722225404403, 51551916515442115079024221439308876243677598340510141

Offset

1,2

Comments

gamma = 1 - 1/12 - 43/420 - 20431/240240 - 2150797323119/36100888223400 - ..., see formula (36) in the reference below.

Links

G. C. Greubel, Table of n, a(n) for n = 1..10

Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in 1/pi^2 and into the formal enveloping series with rational coefficients only. Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. arXiv version, arXiv:1501.00740 [math.NT], 2015.

Formula

a(n) = n * Sum_{k = 2^n + 1 .. 2^(n + 1)} (-1)^(k + 1)/k.

Example

Numerators of 1/12, 43/420, 20431/240240, 2150797323119/36100888223400, ...

Mathematica

a[n_] := Numerator[n*Sum[(-1)^(k + 1)/k, {k, 2^n + 1, 2^(n + 1)}]]; Table[a[n], {n, 1, 8}]

Prog

(PARI) a(n) = numerator(n*sum(k=2^n + 1, 2^(n + 1), (-1)^(k + 1)/k));
(Magma) [Numerator(n*(&+[(-1)^(k+1)/k: k in [2^n+1..2^(n+1)]])): n in [1..6]]; // G. C. Greubel, Oct 28 2018
(GAP) List(List([1..6], n->n*Sum([2^n+1..2^(n+1)], k->(-1)^(k+1)/k)), NumeratorRat); # Muniru A Asiru, Oct 29 2018

Crossrefs

Cf. A075266, A075267, A001620, A195189, A002657, A002790, A262235, A075266, A006953, A001067, A262858 (denominators of this series).

Sequence in context: A361892 A356203 A184144 * A302484 A177488 A262648

Adjacent sequences: A262853 A262854 A262855 * A262857 A262858 A262859

Keyword

frac,nonn

Author

Iaroslav V. Blagouchine, Oct 03 2015

Status

approved