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 1-4 of 4 results.

A382635 Decimal expansion of the multiple prime zeta value p[3, 2].

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Apr 01 2025

Keywords

Comments

Prime multiple zeta constants p[m,...,n] are equivalents of multiple zeta constants when successive natural numbers are replaced by successive primes.
For complete list of multiple prime zeta values up to weight 6 see A382234.

Examples

			0.014095768754803833512720313599879974885...

Crossrefs

Cf. A085548, A085541, A085965, A382234, A382235, A382236, A382634.
Cf. A258983.

Programs

  • Mathematica
    p3 = N[PrimeZetaP[3], 50]; p = 2; sum = 0; sum1 = 0; diff = 0; Monitor[Do[sum = sum + N[1/p^3, 50]; diff = p3 - sum; sum1 = sum1 + diff/p^2; p = NextPrime[p], {n, 1, 100000000}], {sum1, n}]

Formula

Equals Sum_{p,q prime p>q} 1/(p^3*q^2).
Equals A085548*A085541 - A085965 - A382634.
For partial sums and in infinity occurs identity:
p[2, 3] + p[3, 2] + p[2, 1, 2] + p[2, 2, 1] = p[1]*p[2, 2] - p[1, 2, 2]
where p[1] and p[1, 2, 2] are divergent series then
lim_{n->oo} p[1](n)*A382234 - p[1, 2, 2](n) = 0.067101047034256...

A382636 Decimal expansion of the multiple prime zeta value p[2, 1].

Original entry on oeis.org

1, 5, 2, 6, 6, 1, 4, 1, 1, 2, 5, 4, 2
Offset: 0

Views

Author

Artur Jasinski, Apr 07 2025

Keywords

Comments

Prime multiple zeta constants p[m,...,n] are equivalents of multiple zeta constants when successive natural numbers are replaced by successive primes.
For complete list of multiple prime zeta values up to weight 6 see A382234.

Examples

			0.1526614112542...

Crossrefs

Cf. A085548, A085541, A085965, A382234, A382235, A382236, A382634, A382635, A382637.

Programs

  • Mathematica
    p2 = N[PrimeZetaP[2], 50]; p = 2; sum = 0; sum1 = 0; diff = 0; Monitor[Do[sum = sum + N[1/p^2, 50]; diff = p2 - sum; sum1 = sum1 + diff/p; p = NextPrime[p], {n, 1, 100000000}], {sum1, n}]

Formula

Equals p[2, 1] = Sum_{p,q prime p>q} 1/(p^2*q).
Equals p[2, 1] = (p[2, 3] + p[4, 1] + p[2, 1, 2] + 2 p[2, 2, 1])/A085548.
Equals p[2, 1] = sqrt(p[4, 2] + 2 p[2, 2, 2] + 2 p[2, 3, 1] + 2 p[4, 1, 1] + 2 p[2, 1, 2, 1] + 4 p[2, 2, 1, 1]).
A085548*p[2, 1] - p[2, 1, 2] = 0.0531558219243989116479829... [25 digits accuracy].
For partial sums and in infinity occurs identities:
(1) lim_{x->oo} (p[1](x)*A085548 - p[1, 2](x)) = p[2, 1] + A085541 = const.
(2) lim_{x->oo} p[1](x)^3 - 2*p[1](x)*A085548 - p[1, 2](x) - 6*p[1, 1, 1](x) = p[2, 1] - A085541 = const.
(3) lim_{x->oo} (p[1](x)^3 - 3*p[1, 2](x) - 6*p[1, 1, 1](x) = 3*p[2, 1] + A085541 = const.
(4) lim_{x->oo} (p[1](x)^3 - p[1](x)*A085548 - p[1](x)*p[1, 1](x) - p[1, 2](x) - 3*p[1, 1, 1](x)) = p[2, 1] = const.
(5) lim_{x->oo} (p[1](x)*p[1, 1](x) - p[1, 2](x) - 3*p[1, 1, 1](x)) = p[2, 1] = const.
on the left side of each eq. (1)-(5) are divergent series: p[1], p[1, 1], p[1, 2], p[1, 1, 1].

A382637 Decimal expansion of the multiple prime zeta value p[3, 1].

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Apr 27 2025

Keywords

Comments

Prime multiple zeta constants p[m,...,n] are equivalents of multiple zeta constants when successive natural numbers are replaced by successive primes.
For complete list of multiple prime zeta values up to weight 6 see A382234.

Examples

			0.03053116405794...

Crossrefs

Cf. A085548, A085541, A085965, A382234, A382235, A382236, A382634, A382635, A382636, A383432.

Programs

  • Mathematica
    p3 = N[PrimeZetaP[3], 50]; p = 2; sum = 0; sum1 = 0; diff = 0; Monitor[
 Do[sum = sum + N[1/p^3, 50]; diff = p3 - sum; sum1 = sum1 + diff/p;
  p = NextPrime[p], {n, 1, 100000000}], {sum1, n}]
  • PARI
    f(e)=my(S=sumeulerrat(1/x^3), u=0., v=0); forprime(p=2, 2^e, u+=v*S; S-=1/p^3; v=1/p); u;f(30) // Bill Allombert

Formula

Equals Sum_{p,q prime p>q} 1/(p^3*q).
p[3, 1] + p[4] = lim_{x->oo} p[3]*p[1](x) - p[1, 3](x) = 0.1075243038221868... = cons, where p[1] and p[1, 3] are divergent series.
p[3, 1] + p[2, 2] + 2 p[2, 1, 1] = lim_{x->oo} p[2, 1]*p[1](x) - p[1, 2, 1](x) = 0.1686331457227234... = cons, where p[1] and p[2, 1, 1] are divergent series.

A383432 Decimal expansion of the multiple prime zeta value p[2, 1, 1].

Original entry on oeis.org

0, 3, 7, 1, 6, 7, 3, 4, 3, 5, 4
Offset: 0

Views

Author

Artur Jasinski, Apr 27 2025

Keywords

Comments

Prime multiple zeta constants p[m,...,n] are equivalents of multiple zeta constants when successive natural numbers are replaced by successive primes.
For complete list of multiple prime zeta values up to weight 6 see A382234.

Examples

			0.03716734354...

Crossrefs

Cf. A085548, A085541, A085965, A382234, A382235, A382236, A382634, A382635, A382636, A382637.

Programs

  • PARI
    f(e)=my(S=sumeulerrat(1/x^2),u=0.,v=0,w=0.);forprime(p=2,prime(2^e),u+=v*S;S-=1/p^2;v=w/p;w+=1/p);u;
f(30) \\ Bill Allombert

Formula

Equals Sum_{p,q,r prime p>q>r} 1/(p^2*q*r).
Equals (p[2, 1, 3] + p[2, 3, 1] + p[4, 1, 1] + p[2, 1, 1, 2] + p[2, 1, 2, 1] + 2 p[2, 2, 1, 1])/p[2].
Equals (p[2] p[4] + p[2, 1]^2 + p[2] p[2, 2] + p[2, 4] + p[2, 1, 3] + p[2, 2, 2] - p[2, 3, 1] - p[4, 1, 1] + p[2, 1, 1, 2] - p[2, 1, 2, 1] - 2 p[2, 2, 1, 1] - p[2]^3)/p[2].
Showing 1-4 of 4 results.

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