A243350 -id:A243350 - cp's OEIS frontend

A243584 Decimal expansion of 1/(eta*P'(eta)), a constant related to the asymptotic evaluation of the number of prime multiplicative compositions, where eta is A243350, the unique solution of P(x)=1, P being the prime zeta P function (P(x) = sum_(p prime) 1/p^x).

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

Offset: 0

Jean-François Alcover, Jun 06 2014

			0.41277323709367...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.5 Kalmar's composition constant, p. 293.

Eric Weisstein's MathWorld, Prime Zeta function

Cf. A243350.

digits = 30; eta = x /. FindRoot[PrimeZetaP[x] == 1, {x, 7/5},  WorkingPrecision -> digits + 200]; c = N[1/(eta*PrimeZetaP'[eta]) // Re, digits + 200]; RealDigits[c, 10, digits ] // First (* updated Sep 11 2015 *)

A102375 Decimal expansion of reciprocal of the smallest positive zero of sum_{j>0} f(j) where f(j)=[(-1)^(j+1)]*x^(2^(j+1)-2-j)/[(1-x)(1-x^3)(1-x^7)...(1-x^(2^j-1))].

Original entry on oeis.org

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

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 05 2005

S. R. Finch, Kalmar's composition constant, June 5, 2003. [Cached copy, with permission of the author]
Philippe Flajolet and Helmut Prodinger, Level number sequences for trees.

Cf. A243350, A243584.

digits = 103; m0 = 5; dm = 2; Clear[f, g]; f[x_, m_] := Sum[((-1)^(j + 1)*x^( 2^(j + 1) - 2 - j))/Product[1 - x^(2^k - 1), {k, 1, j}] , {j, 1, m}] // N[#, digits]&; g[m_] := g[m] = (1/x /. FindRoot[f[x, m] == 1, {x, 5/9, 4/9, 6/9}, WorkingPrecision -> digits ]); g[m0]; g[m = m0 + dm]; While[RealDigits[g[m], 10, digits] != RealDigits[g[m - dm], 10, digits], Print["m = ", m]; m = m + dm]; RealDigits[g[m], 10, digits] // First (* Jean-François Alcover, Jun 19 2014 *)

1.79414718754168546349846498809380776370136441826513556471412914588110115...

A260623 Decimal expansion of the real solution x to zeta(x) - primezeta(x) = 2.

Original entry on oeis.org

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

Offset: 1

Matthew Campbell, Oct 06 2015

This is also the solution x to Sum_{c composite}(1/c^x) = 1.

			1.4257...

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..100

For the prime analog, see A243350.

x /. FindRoot[Zeta[x] - PrimeZetaP[x] == 2, {x, 3/2}, WorkingPrecision -> 100] // RealDigits // First (* Jean-François Alcover, May 07 2021 *)

solve(x=1.1, 2, zeta(x) - sumeulerrat(1/p, x) - 2) \\ Michel Marcus, May 07 2021

a(6) corrected and a(7)-a(87) added by Hiroaki Yamanouchi, Nov 12 2015

cp's OEIS Frontend

A243584 Decimal expansion of 1/(eta*P'(eta)), a constant related to the asymptotic evaluation of the number of prime multiplicative compositions, where eta is A243350, the unique solution of P(x)=1, P being the prime zeta P function (P(x) = sum_(p prime) 1/p^x).

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A102375 Decimal expansion of reciprocal of the smallest positive zero of sum_{j>0} f(j) where f(j)=[(-1)^(j+1)]*x^(2^(j+1)-2-j)/[(1-x)(1-x^3)(1-x^7)...(1-x^(2^j-1))].

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A260623 Decimal expansion of the real solution x to zeta(x) - primezeta(x) = 2.

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