A136271 -id:A136271 - cp's OEIS frontend

A360095 Decimal expansion of Sum_{p primes, p == 3 (mod 4)} log(p)/p^2.

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

Offset: 0

Vaclav Kotesovec, Jan 25 2023

			0.212444768931665057705067792682825214870373695843766697810497537167709...

X. Gourdon and P. Sebah, Some Constants from Number theory.

Cf. A085548, A085991, A086239, A136271, A358789, A360094.

beta[s_]:= (1 - 1/2^s) * Zeta[s] / DirichletBeta[s]; Do[Print[N[-1/2*Sum[MoebiusMu[2*n + 1]/(2*n + 1) * D[Log[beta[(2*n + 1)*s]], s] /. s->2, {n, 0, m}], 120]], {m, 10, 100, 10}]

Equals A136271 - A360094 - log(2)/4.

A154928 Decimal expansion of Sum_{q in A001358} log(q)/q^2 over the semiprimes q = 4,6,9,...

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 17 2009

Semiprime analog of A136271. The absolute value of the first derivative of the semiprime zeta function at 2.

			Equals 0.0283606815... = log(4)/16 + log(6)/36 + log(9)/81 + ....

R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], 2008-2009; see Table 4 in version 3.

Equals Sum_{j>=1} log(A001358(j))/A074985(j).

Missing zero inserted. Artur Jasinski, Jul 29 2025

A373809 Decimal expansion of the second derivative P''(2) of the prime zeta function at 2.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Aug 18 2024

			0.74159785498280500302394032754503754132161405686935...

Tengiz O. Gogoberidze, Baker's dozen digits of two sums involving reciprocal products of an integer and its greatest prime factor, arXiv:2407.12047 [math.GM], 2024, Table 2, s=2.

Cf. A085548 (P(2)), A136271 (P'(2)).

$MaxExtraPrecision = 200; RealDigits[N[PrimeZetaP''[2], 120]][[1]] (* Amiram Eldar, Aug 19 2024 *)

Equals Sum_{primes = 2,3,5,7,...} log(p)^2/p^2.

More terms from Amiram Eldar, Aug 19 2024

A375509 Decimal expansion of the negated derivative P'(3/2) of the prime zeta function at 3/2.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 18 2024

			1.29571075479160755372288832464555766349416294698452...

Cf. A338574 (P(3/2)), A136271 (P'(2)).

$MaxExtraPrecision = 100; RealDigits[N[PrimeZetaP'[3/2], 120]][[1]] (* Amiram Eldar, Aug 19 2024 *)

P'(3/2) = -Sum_{p = 2,3,5,7,11,..} log(p)/p^(3/2) = -1.29571075479160755372...

cp's OEIS Frontend

A360095 Decimal expansion of Sum_{p primes, p == 3 (mod 4)} log(p)/p^2.

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A154928 Decimal expansion of Sum_{q in A001358} log(q)/q^2 over the semiprimes q = 4,6,9,...

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A373809 Decimal expansion of the second derivative P''(2) of the prime zeta function at 2.

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A375509 Decimal expansion of the negated derivative P'(3/2) of the prime zeta function at 3/2.

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