A393690 - OEIS

A393690

Decimal expansion of lim_{x->oo} Sum_{p<=x} log(p)^2/p - log(x)^2/2 (negated).

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

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OFFSET

1,1

LINKS

Table of n, a(n) for n=1..105.

Artur Kawalec, On the series expansion of the prime zeta function about s = 1 and its coefficients, arXiv:2603.21535 [math.NT], 2026, p. 3 eq. 15 and 17.

FORMULA

Equals gamma^2 + 2*gamma1 - Sum_{n>=2} mu(n)*n*(-Zeta'(n)^2/Zeta(n)^2 + Zeta''(n)/Zeta(n)), where gamma1 is StieltjesGamma(1).

Equals -Integral_{x>=1} (pi(x) - li(x))*(log(x)^2 - 2*log(x))/x^2 dx.

Equals: -2 * 2-th coefficient of expansion of the function P(x)-log(x-1) at x=1, where P(x) is Prime zeta function (coefficient by (x-1)^2).

Equals: -2 * 2-th coefficient of expansion of the function P(1+1/x)-log(x) at x=Infinity, where P(x) is Prime zeta function (coefficient by 1/x^2).

Equals A155969 - A346832 + A394622.

EXAMPLE

2.555107615446445239595583797989464445...

MATHEMATICA

N[EulerGamma^2 + 2*StieltjesGamma[1] - Sum[MoebiusMu[k]*k*(-Zeta'[k]^2/Zeta[k]^2 + Zeta''[k]/Zeta[k]), {k, 2, 1000}], 110] (* Vaclav Kotesovec *)

CROSSREFS

Cf. A083343, A143524, A155969, A346832, A394622.

Sequence in context: A363764 A116698 A246900 * A277086 A229710 A240947

Adjacent sequences: A393687 A393688 A393689 * A393691 A393692 A393693

KEYWORD

nonn,cons

AUTHOR

Artur Jasinski, Mar 25 2026

STATUS

approved