cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A345308 Decimal expansion of Sum_{p primes} log(p) / (p-1)^2.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Jun 13 2021

Keywords

Examples

			1.226968805653470005965662568745762562988257454901426311714794620109...

Links

  • Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 156 (constant C10 + 1).
  • Eric Weisstein's World of Mathematics, Prime Factor, formula (13)-(14), (constant U).

Crossrefs

Cf. A138312, A306016, A345364.

Programs

  • Mathematica
    ratfun = 1/((p - 1)^2); zetas = 0; ratab = Table[konfun = Simplify[ratfun + c/(p^power - 1)] // Together; coefs = CoefficientList[Numerator[konfun], p]; sol = Solve[Last[coefs] == 0, c][[1]]; zetas = zetas + c*Zeta'[power]/Zeta[power] /. sol; ratfun = konfun /. sol, {power, 2, 25}]; Do[Print[N[Sum[Log[p]*ratfun /. p -> Prime[k], {k, 1, m}] + zetas, 110]], {m, 1000, 5000, 1000}]

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

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Apr 27 2025

Keywords

Examples

			0.8844818339635238851965361...

Links

Crossrefs

Cf. A345364.

Programs

  • 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]]
  • 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)

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.

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

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Apr 28 2025

Keywords

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]
Showing 1-3 of 3 results.

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