A086241 -id:A086241 - cp's OEIS frontend

A368644 Decimal expansion of the Mertens constant M(3,2) arising in the formula for the sum of reciprocals of primes p == 2 (mod 3).

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

Offset: 0

Amiram Eldar, Jan 02 2024

Data were taken from Languasco and Zaccagnini's web site.

			0.28505435902375257954174307249854842119682217947187...

Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 204.

Alessandro Languasco and Alessandro Zaccagnini, Computing the Mertens and Meissel-Mertens constants for sums over arithmetic progressions, Experimental Mathematics, Vol. 19, No. 3 (2010), pp. 279-284; arXiv preprint, arXiv:0906.2132 [math.NT], 2009.
Alessandro Languasco and Alessandro Zaccagnini, Computation of the Mertens and Meissel-Mertens constants for sums over arithmetic progressions.

Cf. A001620, A003627, A077761, A086241, A161529, A301429, A368645, A368646.

Equals A086241 - A161529.
Equals lim_{x->oo} (Sum_{primes p == 2 (mod 3), p <= x} 1/p - log(log(x))/2).
Equals gamma/2 - log(sqrt(Pi/3)/(2*K_3)) + Sum_{prime p == 2 (mod 3)} (log(1-1/p) + 1/p), where gamma is Euler's constant (A001620) and K_3 = A301429.

A175642 Decimal expansion of the negated Dirichlet Prime L-function of the real non-principal character mod 5 at 1.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 01 2010

The absolute value of S(1,chi_3) = sum_{primes p = A000040} A080891(p)/p = -1/2 -1/3 -1/7 +1/11 -1/13 -1/17 -1/23 +...

			S(1,chi_3) = -1.0079965479398611722616660755126785669990319566493...

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Cf. A086241 (mod 3), A086239 (mod 4), A175643 (mod 6).

S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums); $MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[-S[5, 3, 1], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 22 2021 *)

More digits from Vaclav Kotesovec, Jan 22 2021

A175643 Decimal expansion of the negated Dirichlet Prime L-function of the real non-principal character mod 6 at 1.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Aug 01 2010

The absolute value of S(1,chi_2) = sum_{primes p = A000040} A134667(p)/p = -1/5 +1/7 -1/11+1/13 -1/17 +1/19 -1/23 +...

			S(1,chi_2) = -0.14194483853319570866139263972173431667541104401...

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Cf. A086241 (mod 3), A086239 (mod 4), A175642 (mod 5).

Do[Print[N[-Log[4/3]/2 + Sum[Log[(Zeta[2*k + 1, 1/6] - Zeta[2*k + 1, 5/6])^2 / ((2^(4*k + 2) - 1) * (3^(4*k + 2) - 1) * Zeta[4*k + 2])] * MoebiusMu[2*k + 1]/(4*k + 2), {k, 1, m}], 120]], {m, 20, 200, 20}] (* Vaclav Kotesovec, Jun 27 2020 *)
S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums); $MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[-S[6, 2, 1], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 22 2021 *)

More terms from Vaclav Kotesovec, Jun 27 2020

A368647 The number of distinct primes of the form 3k+2 dividing n minus the number of distinct primes of the form 3k+1 dividing n.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, -1, 1, 0, 2, 1, 1, -1, 0, 1, 1, 1, 1, -1, 2, -1, 2, 1, 1, 1, 0, 0, 0, 1, 2, -1, 1, 1, 2, 0, 1, -1, 0, -1, 2, 1, 0, -1, 2, 1, 2, 1, 1, -1, 2, 1, 0, 1, 1, 2, 0, -1, 2, 1, 2, -1, 0, -1, 1, 0, 2, -1, 2, 1, 1, 1, 1, -1, 0, 1, 0, 0, 0, -1, 2, 0, 2

Offset: 1

Amiram Eldar, Jan 02 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002476, A003627, A005094, A005088, A005090, A086241.

f[p_, e_] := Switch[Mod[p, 3], 0, 0, 1, -1, 2, 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(p = factor(n)[, 1]); sum(i = 1, #p, if(p[i]%3 == 0, 0, if(p[i]%3 == 1, -1, 1)));}

Additive with a(p^e) = 0 if p = 3, 1 if p == 2 (mod 3), and -1 if p == 1 (mod 3).
a(n) = A005090(n) - A005088(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A086241 = 0.641944... .

A175641 Decimal expansion of the negated Dirichlet Prime L-function of the non-principal character mod 3 at 2.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Aug 01 2010

The absolute value of S(2,chi_2) = sum_{primes p = A000040} A102283(p)/p^2 = -1/2^2 -1/5^2 +1/7^2 -1/11^2 +1/13^2 -1/17^2 +...

			S(2,chi_2) = -0.274705208285518486289577898160860630099...

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Cf. A086241.

S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums); $MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[-S[3, 2, 2], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 22 2021 *)

More digits from Vaclav Kotesovec, Jun 27 2020

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A368644 Decimal expansion of the Mertens constant M(3,2) arising in the formula for the sum of reciprocals of primes p == 2 (mod 3).

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A175642 Decimal expansion of the negated Dirichlet Prime L-function of the real non-principal character mod 5 at 1.

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A175643 Decimal expansion of the negated Dirichlet Prime L-function of the real non-principal character mod 6 at 1.

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A368647 The number of distinct primes of the form 3*k+2 dividing n minus the number of distinct primes of the form 3*k+1 dividing n.

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A175641 Decimal expansion of the negated Dirichlet Prime L-function of the non-principal character mod 3 at 2.

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Mathematica

Extensions

A368647 The number of distinct primes of the form 3k+2 dividing n minus the number of distinct primes of the form 3k+1 dividing n.