A397227 - OEIS

A397227

Decimal expansion of Sum_{k>=1} (-1)^(k+1) * AH(k)/k^6, where AH(k) = A058313(k)/A058312(k) is the k-th alternating harmonic (or skew-harmonic) number.

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

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OFFSET

0,1

LINKS

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

Ali Shadhar Olaikhan, An Introduction to the Harmonic Series and Logarithmic Integrals, 2nd ed., 2023, section 4.1, p. 261.

Cornel Ioan Vălean, More (Almost) Impossible Integrals, Sums, and Series, Springer Cham, 2023. See section 4.22, p. 427.

Index entries for sequences related to harmonic numbers.

FORMULA

Equals 63*log(2)*zeta(6)/32 + 15*zeta(2)*zeta(5)/16 + 3*zeta(3)*zeta(4)/4 - 377*zeta(7)/128.

EXAMPLE

0.993230217632557700220529966868623945355529352096714...

MATHEMATICA

RealDigits[63*Log[2]*Zeta[6]/32 + 15*Zeta[2]*Zeta[5]/16 + 3*Zeta[3]*Zeta[4]/4 - 377*Zeta[7]/128, 10, 120][[1]]

PROG

(PARI) 63*log(2)*zeta(6)/32 + 15*zeta(2)*zeta(5)/16 + 3*zeta(3)*zeta(4)/4 - 377*zeta(7)/128

CROSSREFS

Cf. A058312, A058313.

Cf. A002117, A002162, A013661, A013662, A013663, A013664, A013665.

Sum_{k>=1} (-1)^(k+1) * AH(k)/k^m: A369884 (m=1), A397225 (m=2), A397226 (m=4), this constant (m=6), A397228 (m=8).

Sequence in context: A254273 A011211 A074879 * A387447 A291366 A146490

Adjacent sequences: A397224 A397225 A397226 * A397228 A397229 A397230

KEYWORD

nonn,cons

AUTHOR

Amiram Eldar, Jun 19 2026

STATUS

approved