A383486 Decimal expansion Sum_{p primes} (p^2 + p^4)*log(p)^3/(p^6 - 3*p^4 + 3*p^2 -1).

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

Offset: 1

Author

Artur Jasinski, Apr 28 2025

Keywords

• nonn
• cons

Examples

			1.950135832673189957954522125256874...
		

Crossrefs

Cf. A345364, A383224.

Programs

Mathematica

RealDigits[-(Zeta'''[2]*Zeta[2]^2 - 3*Zeta''[2]*Zeta'[2]*Zeta[2] +
  2*Zeta'[2]^3)/Zeta[2]^3, 10, 105][[1]]
(* Sum_{primes p} f[p]*log[p]^elog, elog > 0 *) $MaxExtraPrecision = 1000; Clear[f]; f[p_] := (p^2 + p^4)/(p^6 - 3*p^4 + 3*p^2 - 1); elog = 3; Do[cc = Rest[CoefficientList[Series[f[1/x], {x, 0, m}], x, m + 1]]; Print[Sum[Log[Prime[k]]^elog*f[Prime[k]], {k, 1, 100}] + N[Sum[Indexed[cc, n]*((-1)^elog*Derivative[elog][PrimeZetaP][n] - Sum[Log[Prime[k]]^elog/Prime[k]^n, {k, 1, 100}]), {n, 2, m}], 110]], {m, 100, 500, 100}] (* Vaclav Kotesovec, Apr 28 2025 *)

PARI

/* procedure by Bill Allombert * /
/* this version requires PARI 2.18.1 and up */
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)*s^d
    ,d==0,
    sumeulerrat(f,s,a)
    ,d>0,
    my(prec=getlocalbitprec(),F=f);
    f = subst(f,p,1/p)+O(p^prec);
    for(i=1,d, f=intformal(f/p));
    f = truncate(f);
    my(t=0, N=max(a, ceil((2^prec*normlp(f))^(1/(poldegree(f)*s)))));
    forprime(l=a,N-1,t+=subst(F,p,l^s)*log(l)^d);
    t+(-1)^d*derivnum(t=1,sumeulerrat(subst(f,p,1/p),t*s,N),d)/s^d);
}
SumEulerLog( (p^2+p)/(p^3-3*p^2+3*p-1),2,,3)

Formula

Equals (3*zeta''(2)*zeta'(2)*zeta(2) - zeta'''(2)*zeta(2)^2 - 2*zeta'(2)^3)/zeta(2)^3. [formula found by Bill Allombert]