A383224 Decimal expansion Sum_{p primes} log(p)^2*p^2/(p^2-1)^2.

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

Offset

0,1

Links

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

Bill Allombert, SumEulerLog procedure, Pari gp procedures.

Formula

Equals 6*(Pi^2*zeta''(2)-6*zeta'(2)^2)/Pi^4.

Equals 6*(Pi^2*zeta''(2)-6*(zeta[2]*(gamma + log(2*Pi) - 12*log(A)))^2)/Pi^4 where A is Glaisher-Kinkelin constant A074962.

Equals zeta''(2)/zeta(2)-zeta'(2)^2/zeta(2)^2 see A201994, A073002 and A013661.

Example

0.8844818339635238851965361...

Maple

Zeta(2, 2)/Zeta(2) -Zeta(1, 2)^2/Zeta(2)^2 ; evalf(%) ; # R. J. Mathar, May 07 2025

Mathematica

RealDigits[(6 (-6 Zeta'[2]^2 + Pi^2 Zeta''[2]))/Pi^4, 10, 105][[1]]

Prog

(PARI)
/* Procedure by Bill Allombert */
default(realprecision, 105);
SumEulerLog(f, s=1, a=2, d=1)=
{
  my(p=variable(f));
  if(type(d)!="t_INT", error("incorrect type in SumEulerLog"));
  if (d<0,
    d=-d;
    for(i=1, d, f=deriv(f)*p);
    (-1)^d*intnum(t=1, [oo, log(2)*s], (t-1)^(d-1)*sumeulerrat(f, t*s, a))/gamma(d)
    , d==0,
    sumeulerrat(f, s, a)
    , d>0,
    my(S=0, v);
    my(prec=getlocalbitprec());
    f=subst(f, 'p, 1/p)+O(p^prec);
    for(i=1, d, f=intformal(f/p));
    v = valuation(f, p);
    f = truncate(f);
    for(i=v, prec/(v-1),
     S += polcoef(f, i)*derivnum(t=1, sumeulerrat(1/p, t*i*s, a), d));
    (-1)^d*S);
}
SumEulerLog(p^2/(p^2-1)^2, , , 2)

Crossrefs

Cf. A345364.

Sequence in context: A085669 A154012 A154845 * A370971 A126600 A388192

Adjacent sequences: A383221 A383222 A383223 * A383225 A383226 A383227

Keyword

nonn,cons

Author

Artur Jasinski, Apr 27 2025

Status

approved