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

%I A383486 #20 Apr 29 2025 04:52:08
%S A383486 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,
%T A383486 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,
%U A383486 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
%N A383486 Decimal expansion Sum_{p primes} (p^2 + p^4)*log(p)^3/(p^6 - 3*p^4 + 3*p^2 -1).
%F A383486 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]
%e A383486 1.950135832673189957954522125256874...
%t A383486 RealDigits[-(Zeta'''[2]*Zeta[2]^2 - 3*Zeta''[2]*Zeta'[2]*Zeta[2] +
%t A383486   2*Zeta'[2]^3)/Zeta[2]^3, 10, 105][[1]]
%t A383486 (* 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 *)
%o A383486 (PARI)
%o A383486 /* procedure by Bill Allombert * /
%o A383486 /* this version requires PARI 2.18.1 and up */
%o A383486 SumEulerLog(f,s=1,a=2,d=1)=
%o A383486 {
%o A383486   my(p=variable(f));
%o A383486   if(type(d)!="t_INT",error("incorrect type in SumEulerLog"));
%o A383486   if (d<0,
%o A383486     d=-d;
%o A383486     for(i=1,d, f=deriv(f)*p);
%o A383486     (-1)^d*intnum(t=1,[oo,log(2)*s],(t-1)^(d-1)*sumeulerrat(f,t*s,a))/gamma(d)*s^d
%o A383486     ,d==0,
%o A383486     sumeulerrat(f,s,a)
%o A383486     ,d>0,
%o A383486     my(prec=getlocalbitprec(),F=f);
%o A383486     f = subst(f,p,1/p)+O(p^prec);
%o A383486     for(i=1,d, f=intformal(f/p));
%o A383486     f = truncate(f);
%o A383486     my(t=0, N=max(a, ceil((2^prec*normlp(f))^(1/(poldegree(f)*s)))));
%o A383486     forprime(l=a,N-1,t+=subst(F,p,l^s)*log(l)^d);
%o A383486     t+(-1)^d*derivnum(t=1,sumeulerrat(subst(f,p,1/p),t*s,N),d)/s^d);
%o A383486 }
%o A383486 SumEulerLog( (p^2+p)/(p^3-3*p^2+3*p-1),2,,3)
%Y A383486 Cf. A345364, A383224.
%K A383486 nonn,cons
%O A383486 1,2
%A A383486 _Artur Jasinski_, Apr 28 2025