A335089 Decimal expansion of log(Pi^2/6).

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

Offset

0,1

Links

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

Mathematics Stack Exchange, Infinite series involving Von Mangoldt's function.

Grant Sanderson, What makes the natural log "natural"?, 3Blue1Brown video (2020).

Eric Weisstein's World of Mathematics, Mangold Function.

Eric Weisstein's World of Mathematics, Prime Zeta Function.

Formula

Equals Sum_{k>=2} MangoldtLambda(k) / ((k^2)*log(k)).

Equals Sum_{k>=1} (1/k)*(1/(A246655(n)^2)) where k is the exponent of the prime power, A025474(n+1).

Equals Sum_{k>=1} primezeta(2*k)/k.

Equals 2*log(Pi) - log(6).

Equals log(zeta(2)) = log(A013661).

Example

Equals 1/(2^2) + 1/(3^2) + (1/(4^2))*(1/2) + 1/(5^2) + + 1/(7^2) + (1/(8^2))*(1/3) + ... = 0.4977003024707...

Mathematica

RealDigits[Log[Pi^2/6], 10, 120][[1]]
RealDigits[Sum[PrimeZetaP[2 k]/k, {k, 1, inf}], 10, 120][[1]]

Prog

(PARI) log(Pi^2/6) \\ Michel Marcus, Sep 15 2020

Crossrefs

Cf. A000796, A013661, A053510, A131659, A016629.

Cf. A100995, A014963, A025474, A246655.

Sequence in context: A166923 A021205 A296425 * A306004 A056992 A339023

Adjacent sequences: A335086 A335087 A335088 * A335090 A335091 A335092

Keyword

nonn,cons

Author

Terry D. Grant, Sep 11 2020

Status

approved