A398020 Decimal expansion of -Sum_{k>=1} lambda(k) * theta(k) * log(k) / k^2, where lambda is Liouville's function (A008836), and theta(k) is the number of unitary divisors of k (A034444).

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

Offset

0,1

Links

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

Matthew Perkins and Robert A. Van Gorder, Closed-form calculation of infinite products of Glaisher-type related to Dirichlet series, The Ramanujan Journal, Vol. 49, No. 2 (2019), pp. 371-389. See p. 382, Corollary 2.11.

Formula

Equals -Sum_{k>=1} A158522(k) * log(k) / k^2.

Equals (48/5)*log(A) + 96*log(A_3), where A is the Glaisher-Kinkelin constant (A074962), and A_3 is the third Bendersky constant (A243263).

Note that Sum_{k>=1} lambda(k) * theta(k) / k^2 = 2/5.

Example

0.4050329825113293439229325474642985858842963692909017...

Mathematica

RealDigits[(48/5)*Log[Glaisher] - 96*Zeta'[-3] - 22/15, 10, 120][[1]]

Prog

(PARI) -(48/5)*zeta'(-1) - 96*zeta'(-3) - 2/3

Crossrefs

Cf. A008836, A034444, A073002 (Sum_{k>=1} log(k)/k^2), A074962, A158522, A243263, A398012, A398019.

Sequence in context: A055241 A055242 A200295 * A352167 A089389 A046779

Adjacent sequences: A398017 A398018 A398019 * A398023 A398024 A398025

Keyword

nonn,cons,new

Author

Amiram Eldar, Jul 17 2026

Status

approved