A343612 -id:A343612 - cp's OEIS frontend

A086032 Decimal expansion of the prime zeta modulo function at 2 for primes of the form 4k+1.

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

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 07 2003

			0.053813763574057670280678287341536562285675501495085532293911422295866827...

Jean-François Alcover, Table of n, a(n) for n = 0..1004
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, section 3.2, constant P(m=4,n=1,s=2).
X. Gourdon and P. Sebah, Some Constants from Number theory.
OEIS index to entries related to the (prime) zeta function.

Cf. A085991 (same for primes 4k+3), A175644 (for primes 3k+1), A343612 (for primes 3k+2), A086033 - A086039 (for 1/p^3, ..., 1/p^9), A085548 (PrimeZeta(2)), A002144 (primes 4k+1).

digits = 1004; nmax0 = 100; dnmax = 10;
Clear[PrimeZeta41];
f[s_] := (1 + 2^-s)^-1*DirichletBeta[s] Zeta[s]/Zeta[2s];
PrimeZeta41[s_, nmax_] := PrimeZeta41[s, nmax] = (1/2) Sum[MoebiusMu[2n + 1]* Log[f[(2n + 1)*2]]/(2n + 1), {n, 0, nmax}] // N[#, digits+5]&;
PrimeZeta41[2, nmax = nmax0];
PrimeZeta41[2, nmax += dnmax];
While[Abs[PrimeZeta41[2, nmax] - PrimeZeta41[2, nmax - dnmax]] > 10^-(digits + 5), Print["nmax = ", nmax]; nmax += dnmax];
PrimeZeta41[2] = PrimeZeta41[2, nmax];
Join[{0}, RealDigits[PrimeZeta41[2], 10, digits][[1]]] (* Jean-François Alcover, Jun 24 2011, after X. Gourdon and P. Sebah, updated May 06 2021 *)

PrimeZeta41(s)={suminf(n=0, my(t=s+s*n*2); moebius(2*n+1)*log(zeta(t)/zeta(2*t)*(zetahurwitz(t,1/4)-zetahurwitz(t,3/4))/(4^t+2^t))/(4*n+2))}
A086032_upto(N=100)={localprec(N+3);digits((PrimeZeta41(2)+1)\.1^N)[^1]} \\  M. F. Hasler, Apr 24 2021

Zeta_Q(2) = Sum_{p in A002144} 1/p^2, where A002144 = {primes p == 1 (mod 4)}.
Equals A085548 - 1/4 - A085991. - R. J. Mathar, Apr 03 2011
Zeta_Q(2) = Sum_{odd m > 0} mu(m)/2m * log(DirichletBeta(2m)*zeta(2m)/zeta(4m)/(1 + 4^-m)) [using Gourdon & Sebah, Theorem 11]. - M. F. Hasler, Apr 26 2021

A175644 Decimal expansion of the sum 1/p^2 over primes p == 1 (mod 3).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Aug 01 2010

The prime zeta modulo function at 2 for primes of the form 3k+1, which is P_{3,2}(2) = Sum_{p in A002476} 1/p^2 = 1/7^2 + 1/13^2 + 1/19^2 + 1/31^2 + ...

The complementary Sum_{p in A003627} 1/p^2 is given by P_{3,2}(2) = A085548 - 1/3^2 - (this value here) = 0.307920758607736436842505075940... = A343612.

			P_{3,1}(2) = 0.03321555032221795055292717778013809648108756665...

Jean-François Alcover, Table of n, a(n) for n = 0..1003
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.
OEIS index to entries related to the (prime) zeta function.

Cf. A086032 (P_{4,1}(2): same for p==1 (mod 4)), A175645 (P_{3,1}(3): same for 1/p^3), A343612 (P_{3,2}(2): same for p==2 (mod 3)), A085548 (PrimeZeta(2)).

With[{s=2}, Do[Print[N[1/2 * Sum[(MoebiusMu[2*n + 1]/(2*n + 1)) * Log[(Zeta[s + 2*n*s]*(Zeta[s + 2*n*s, 1/6] - Zeta[s + 2*n*s, 5/6])) / ((1 + 2^(s + 2*n*s))*(1 + 3^(s + 2*n*s)) * Zeta[2*(1 + 2*n)*s])], {n, 0, m}], 120]], {m, 100, 500, 100}]] (* Vaclav Kotesovec, Jan 13 2021 *)
digits = 1003;
m = 100; (* initial value of n beyond which summand is considered negligible *)
dm = 100; (* increment of m *)
P[s_, m_] (* "P" short name for "PrimeZeta31" *):= P[s, m] = Sum[Module[{t}, t = s + 2 n*s; MoebiusMu[2n + 1]* ((1/(4n + 2)) (-Log[1 + 2^t] - Log[1 + 3^t] + Log[Zeta[t]] - Log[Zeta[2t]] + Log[Zeta[t, 1/6] - Zeta[t, 5/6]]))], {n, 0, m}] // N[#, digits+10]&;
P[2, m]; P[2, m += dm];
While[ RealDigits[P[2,    m]][[1]][[1;;digits]] !=
       RealDigits[P[2, m-dm]][[1]][[1;;digits]], Print["m = ", m]; m+=dm];
Join[{0}, RealDigits[P[2, m]][[1]][[1;;digits]]] (* Jean-François Alcover, Jun 24 2022, after Vaclav Kotesovec *)

my(s=0); forprimestep(p=1, 1e8, 3, s+=1./p^2); s \\ For illustration: primes up to 10^N give only about 2N+2 (= 18 for N=8) correct digits. - M. F. Hasler, Apr 23 2021

PrimeZeta31(s)=suminf(n=0,my(t=s+2*n*s); moebius(2*n+1)*log((zeta(t)*(zetahurwitz(t,1/6)-zetahurwitz(t,5/6)))/((1+2^t)*(1+3^t)*zeta(2*t)))/(4*n+2)) \\ Inspired from Kotesovec's Mmca code
A175644_upto(N=100)={localprec(N+5);digits((PrimeZeta31(2)+1)\.1^N)[^1]} \\ M. F. Hasler, Apr 23 2021

More digits from Vaclav Kotesovec, Jun 27 2020

A343619 Decimal expansion of P_{3,2}(9) = Sum 1/p^9 over primes == 2 (mod 3).

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 25 2021

The prime zeta modulo function P_{m,r}(s) = Sum_{primes p == r (mod m)} 1/p^s generalizes the prime zeta function P(s) = Sum_{primes p} 1/p^s.

			0.0019536374331587137208046015123929176069335003912220646291626134042468494...

Jean-François Alcover, Table of n, a(n) for n = 0..1005
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, value P(m=3, n=2, s=9), p. 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A003627 (primes 3k-1), A001017 (n^9), A085969 (PrimeZeta(9)).
Cf. A343612 - A343618 (P_{3,2}(s): analog for 1/p^s, s = 2 .. 8).
Cf. A343629 (for primes 3k+1), A086039 (for primes 4k+1), A085998 (for primes 4k+3).

digits = 1004; nmax0 = 50; dnmax = 10;
Clear[PrimeZeta31];
PrimeZeta31[s_, nmax_] := PrimeZeta31[s, nmax] = Sum[Module[{t}, t = s + 2 n*s; MoebiusMu[2 n + 1] ((1/(4 n + 2)) (-Log[1 + 2^t] - Log[1 + 3^t] + Log[Zeta[t]] - Log[Zeta[2 t]] + Log[Zeta[t, 1/6] - Zeta[t, 5/6]]))], {n, 0, nmax}] // N[#, digits + 5] &;
PrimeZeta31[9, nmax = nmax0];
PrimeZeta31[9, nmax += dnmax];
While[Abs[PrimeZeta31[9, nmax] - PrimeZeta31[9, nmax - dnmax]] > 10^-(digits + 5), Print["nmax = ", nmax]; nmax += dnmax];
PrimeZeta32[9] = PrimeZetaP[9] - 1/3^9 - PrimeZeta31[9, nmax];
Join[{0, 0}, RealDigits[PrimeZeta32[9], 10, digits][[1]] ] (* Jean-François Alcover, May 07 2021, after M. F. Hasler's PARI code *)

A343619_upto(N=100)={localprec(N+5); digits((PrimeZeta32(9)+1)\.1^N)[^1]} \\ see A343612 for the function PrimeZeta32

P_{3,2}(9) = Sum_{p in A003627} 1/p^9 = P(9) - 1/3^9 - P_{3,1}(9).

A343613 Decimal expansion of P_{3,2}(3) = Sum 1/p^3 over primes == 2 (mod 3).

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 22 2021

The prime zeta modulo function P_{m,r}(s) = Sum_{primes p == r (mod m)} 1/p^s generalizes the prime zeta function P(s) = Sum_{primes p} 1/p^s.

			0.134125178915463540428599329999431198995879919752168337370599106153853349956...

Jean-François Alcover, Table of n, a(n) for n = 0..1006
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, value P(m=3, n=2, s=3) on p. 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A003627 (primes 3k-1), A085541 (PrimeZeta(3)), A021031 (1/27).

Cf. A175645 (same for p==1 (mod 3)), A086033 (for primes 4k+1), A085992 (for primes 4k+3), A343612 - A343619 (P_{3,2}(2..9): same for 1/p^2, ..., 1/p^9).

s=0;forprimestep(p=2,1e8,3,s+=1./p^3);s \\ For illustration: using primes up to 10^N gives about 2N+2 (= 18 for N=8) correct digits.

A343613_upto(N=100)={localprec(N+5); digits((PrimeZeta32(3)+1)\.1^N)[^1]} \\ see A343612 for the function PrimeZeta32.

P_{3,2}(3) = P(3) - 1/3^3 - P_{3,1}(3) = A085541 - A021031 - A175645.

A343614 Decimal expansion of P_{3,2}(4) = Sum 1/p^4 over primes == 2 (mod 3).

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 22 2021

The prime zeta modulo function P_{m,r}(s) = Sum_{primes p == r (mod m)} 1/p^s generalizes the prime zeta function P(s) = Sum_{primes p} 1/p^s.

			P_{3,2}(4) = 0.06418614569655777899009908658740273681...

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.
OEIS index to entries related to the (prime) zeta function.

Cf. A003627 (primes 3k-1), A085964 (PrimeZeta(4)), A021085 (1/3^4).

Cf. A343624 (same for primes 3k+1), A086034 (for primes 4k+1), A085993 (for primes 4k+3), A343612 - A343619 (P_{3,2}(2..9): same for 1/p^2, ..., 1/p^9).

s=0;forprimestep(p=2,1e8,3,s+=1./p^4);s \\ For illustration: using primes up to 10^N gives about 3N+2 (= 26 for N=8) correct digits.

A343614_upto(N=100)={localprec(N+5); digits((PrimeZeta32(4)+1)\.1^N)[^1]} \\ see for the function PrimeZeta32.

P_{3,2}(4) = P(4) - 1/3^4 - P_{3,1}(4) = A085964 - A021085 - A343624.

A343615 Decimal expansion of P_{3,2}(5) = Sum 1/p^5 over primes == 2 (mod 3).

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 22 2021

The prime zeta modulo function P_{m,r}(s) = Sum_{primes p == r (mod m)} 1/p^s generalizes the prime zeta function P(s) = Sum_{primes p} 1/p^s.

			0.0315771357190039419560337803437163963477729963832486145790258341228297557...

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, value P(m=3, n=2, s=5) on p. 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A003627 (primes 3k-1), A000584 (n^5), A085965 (PrimeZeta(5)), A021247 (1/3^5).

Cf. A343625 (same for primes 3k+1), A086035 (for primes 4k+1), A085994 (for primes 4k+3), A343612 - A343619 (P_{3,2}(2..9): same for 1/p^2, ..., 1/p^9).

s=0;forprimestep(p=2,1e8,3,s+=1./p^4);s \\ For illustration: using primes up to 10^N gives about 3N+2 (= 26 for N=8) correct digits.

A343615_upto(N=100, s=5)={localprec(N+5); digits((sumeulerrat(1/p^s)-1/3^s-PrimeZeta31(s)+1)\.1^N)[^1]} \\ see A175644 for the function PrimeZeta31, A343612 for a function PrimeZeta32

P_{3,2}(5) = P(5) - 1/3^5 - P_{3,1}(5).

A343616 Decimal expansion of P_{3,2}(6) = Sum 1/p^6 over primes == 2 (mod 3).

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 25 2021

The prime zeta modulo function P_{m,r}(s) = Sum_{primes p == r (mod m)} 1/p^s generalizes the prime zeta function P(s) = Sum_{primes p} 1/p^s.

			0.015689614727130461563527666152209091814208675553077763366153188676457...

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, value P(m=3, n=2, s=6), p. 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A003627 (primes 3k-1), A001014 (n^6), A085966 (PrimeZeta(6)), A021733 (1/3^6).
Cf. A343612 - A343619 (P_{3,2}(s): analog for 1/p^s, s = 2 .. 9).
Cf. A343626 (for primes 3k+1), A086036 (for primes 4k+1), A085995 (for primes 4k+3).

A343616_upto(N=100)={localprec(N+5); digits((PrimeZeta32(6)+1)\.1^N)[^1]} \\ see A343612 for the function PrimeZeta32

P_{3,2}(6) = Sum_{p in A003627} 1/p^6 = P(6) - 1/3^6 - P_{3,1}(6).

A343618 Decimal expansion of P_{3,2}(8) = Sum 1/p^8 over primes == 2 (mod 3).

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 25 2021

The prime zeta modulo function P_{m,r}(s) = Sum_{primes p == r (mod m)} 1/p^s generalizes the prime zeta function P(s) = Sum_{primes p} 1/p^s.

			0.003908814823388594971406115663072323981226161069324694978359864189332...

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, value P(m=3, n=2, s=8), p. 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A003627 (primes 3k-1), A001016 (n^8), A085968 (PrimeZeta(8)).
Cf. A343612 - A343619 (P_{3,2}(s): analog for 1/p^s, s = 2 .. 9).
Cf. A343628 (for primes 3k+1), A086038 (for primes 4k+1), A085997 (for primes 4k+3).

A343618_upto(N=100)={localprec(N+5); digits((PrimeZeta32(8)+1)\.1^N)[^1]} \\ see A343612 for the function PrimeZeta32

P_{3,2}(8) = Sum_{p in A003627} 1/p^8 = P(8) - 1/3^8 - P_{3,1}(8).

A343617 Decimal expansion of P_{3,2}(7) = Sum 1/p^7 over primes == 2 (mod 3).

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 25 2021

The prime zeta modulo function P_{m,r}(s) = Sum_{primes p == r (mod m)} 1/p^s generalizes the prime zeta function P(s) = Sum_{primes p} 1/p^s.

			0.0078253541130504928742517016707559206033079309751324433146804883394...

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, value P(m=3, n=2, s=7), p. 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A003627 (primes 3k-1), A001015 (n^7), A085967 (PrimeZeta(7)).
Cf. A343612 - A343619 (P_{3,2}(s): analog for 1/p^s, s = 2 .. 9).
Cf. A343627 (for primes 3k+1), A086037 (for primes 4k+1), A085996 (for primes 4k+3).

A343617_upto(N=100)={localprec(N+5); digits((PrimeZeta32(7)+1)\.1^N)[^1]} \\ see A343612 for the function PrimeZeta32

P_{3,2}(7) = Sum_{p in A003627} 1/p^7 = P(7) - 1/3^7 - P_{3,1}(7).

cp's OEIS Frontend

A086032 Decimal expansion of the prime zeta modulo function at 2 for primes of the form 4k+1.

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A175644 Decimal expansion of the sum 1/p^2 over primes p == 1 (mod 3).

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A343619 Decimal expansion of P_{3,2}(9) = Sum 1/p^9 over primes == 2 (mod 3).

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A343613 Decimal expansion of P_{3,2}(3) = Sum 1/p^3 over primes == 2 (mod 3).

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A343614 Decimal expansion of P_{3,2}(4) = Sum 1/p^4 over primes == 2 (mod 3).

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A343615 Decimal expansion of P_{3,2}(5) = Sum 1/p^5 over primes == 2 (mod 3).

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A343616 Decimal expansion of P_{3,2}(6) = Sum 1/p^6 over primes == 2 (mod 3).

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