A382554 -id:A382554 - cp's OEIS frontend

A086242 Decimal expansion of the sum of 1/(p-1)^2 over all primes p.

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

Offset: 1

Eric W. Weisstein, Jul 13 2003

			1.37506499474863528791725313052243969917959996017...

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, 2003, pp. 94-98.

Henri Cohen, High precision computation of Hardy-Littlewood Constants (dvi), 1998.
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
Rafael Jakimczuk, On Sums of Powers of the p-adic Valuation of n!, Journal of Integer Sequences, Vol. 20 (2017), Article 17.5.6.
Eric Weisstein's World of Mathematics, Prime Factor.
Eric Weisstein's World of Mathematics, Prime Sums.
Index to constants which are prime zeta sums {0,2,0}

Cf. A000010, A007434, A119686, A334746, A382554.

digits = 116; Np = NSum[(n-1)*PrimeZetaP[n], {n, 2, Infinity}, NSumTerms -> 3*digits, WorkingPrecision -> digits+10]; RealDigits[Np, 10, digits] // First (* Jean-François Alcover, Sep 02 2015 *)

default(realprecision,256);
(f(k)=return(sum(n=1,1024,moebius(n)/n*log(zeta(k*n)))));
sum(k=2,1024,(k-1)*f(k)) /* Robert Gerbicz, Sep 12 2012 */

sumeulerrat(1/(p-1)^2) \\ Amiram Eldar, Mar 19 2021

Equals Sum_{k>=2} (k-1)*primezeta(k). - Robert Gerbicz, Sep 12 2012
Equals lim_{n -> oo} A119686(n)/A334746(n). - Petros Hadjicostas, May 11 2020
Equals Sum_{k>=2} (J_2(k)-phi(k)) * log(zeta(k)) / k, where J_2 = A007434 and phi = A000010 (Jakimczuk, 2017). - Amiram Eldar, Mar 18 2024

More digits copied from Cohen's paper by R. J. Mathar, Dec 05 2008

More terms from Robert Gerbicz, Sep 12 2012

A382555 Decimal expansion of Sum_{p prime} 1/(p + 1)^3.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.0607162964871363728473985988280913247203267...

Index to constants which are prime zeta sums {0,0,3}.

Cf. A380840, A382554.

sumeulerrat(1/(p+1)^3) \\ Amiram Eldar, Apr 02 2025

Equals Sum_{k>=3} (-1)^(k+1) * ((k-2)*(k-1)/2) * P(k), where P is the prime zeta function. - Amiram Eldar, Apr 02 2025

A324833 Decimal expansion of eta_2, a constant related to the asymptotic density of certain sets of residues.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Mar 17 2019

			0.12903892589780755649743486348177587763849321419920568830041270456398...

Carl Pomerance, Andrzej Schinzel, Multiplicative Properties of Sets of Residues, Moscow Journal of Combinatorics and Number Theory. 2011. Vol. 1. Iss. 1. pp. 52-66. See p. 62.
Index to constants which are prime zeta sums {0,2,2}

Cf. A154945 (eta_1), A324834 (eta_3), A324835 (eta_4), A324836 (eta_5).

digits = 101; m0 = 2 digits; Clear[rd]; rd[m_] := rd[m] = RealDigits[eta2 = Sum[n PrimeZetaP[2n + 2], {n, 1, m}], 10, digits][[1]]; rd[m0]; rd[m = 2m0]; While[rd[m] != rd[m-m0], Print[m]; m = m+m0]; Print[N[eta2, digits] ]; rd[m]

Sum_{p prime} 1/(p^2-1)^2.
Sum_{n>0} n P(2n+2) where P is the prime zeta P function.
Equals - A136141/4 + A086242/4 - A179119/4 + A382554/4. - Artur Jasinski, Mar 31 2025

A324834 Decimal expansion of eta_3, a constant related to the asymptotic density of certain sets of residues.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Mar 17 2019

			0.03907240573557479188765950332042297638668483824477336035675406603269...

Carl Pomerance, Andrzej Schinzel, Multiplicative Properties of Sets of Residues, Moscow Journal of Combinatorics and Number Theory. 2011. Vol. 1. Iss. 1. pp. 52-66. See p. 62.
Index to constants which are prime zeta sums {0,3,3}

Cf. A154945 (eta_1), A324833 (eta_2), A324835 (eta_4), A324836 (eta_5).

digits = 102; m0 = 2 digits; Clear[rd]; rd[m_] := rd[m] = RealDigits[eta3 = Sum[n (n+1)/2 PrimeZetaP[2 n + 4], {n, 1, m}] , 10, digits][[1]]; rd[m0]; rd[m = 2 m0]; While[rd[m] != rd[m-m0], Print[m]; m = m+m0]; Print[N[eta3, digits]]; rd[m]

Sum_{p prime} 1/(p^2-1)^3.
Sum_{n>0} (n(n+1)/2) P(2n+4) where P is the prime zeta P function.
Equals 3*A136141/16 - 3*A086242/16 + A380840/8 + 3*A179119/16 - 3*A382554/16 - A382555/8. - Artur Jasinski, Mar 31 2025

A110833 a(n) = (prime(n)+1)^2.

Original entry on oeis.org

9, 16, 36, 64, 144, 196, 324, 400, 576, 900, 1024, 1444, 1764, 1936, 2304, 2916, 3600, 3844, 4624, 5184, 5476, 6400, 7056, 8100, 9604, 10404, 10816, 11664, 12100, 12996, 16384, 17424, 19044, 19600, 22500, 23104, 24964, 26896, 28224, 30276, 32400, 33124, 36864

Offset: 1

Giovanni Teofilatto, Sep 18 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A008864, A065472, A065486.

[(p+1)^2: p in PrimesUpTo(200)]; // Vincenzo Librandi, Mar 27 2014

Table[(Prime[n] + 1)^2, {n, 200}] (* Vincenzo Librandi, Mar 27 2014 *)

from sympy import primerange
print([(p+1)**2 for p in primerange(1, 192)]) # Michael S. Branicky, Sep 16 2021

From Amiram Eldar, Jan 23 2021: (Start)
a(n) = A008864(n)^2.
Product_{n>=1} (1 + 1/a(n)) = A065486.
Product_{n>=1} (1 - 1/a(n)) = A065472. (End)
Sum 1/a(n) = A382554. - R. J. Mathar, Mar 31 2025

Corrected and extended by Ray Chandler, Oct 08 2005

A382567 Decimal expansion of Sum_{p prime} 1/(p*(p + 1)^2).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.0857526221076099340631462167393937930068857672953086571965324502786040217286...

Index to constants which are prime zeta sums {1,0,2}.

Cf. A179119, A382554.

sumeulerrat(1/(p*(p+1)^2)) \\ Amiram Eldar, Apr 01 2025

Equals A179119 - A382554.

Equals Sum_{k>=3} (-1)^(k+1) * (k-2) * P(k), where P is the prime zeta function. - Amiram Eldar, Apr 01 2025

A382568 Decimal expansion of Sum_{p prime} 1/(p^2*(p + 1)^2).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.03626487166925232342830256000610938052208615647051330838965787931013...

Index to constants which are prime zeta sums {2,0,2}.

Cf. A085548, A179119, A382554.

sumeulerrat(1/(p^2*(p+1)^2)) \\ Amiram Eldar, Apr 01 2025

Equals A085548 - 2*A179119 + A382554.

Equals Sum_{k>=4} (-1)^k * (k-3) * P(k), where P is the prime zeta function. - Amiram Eldar, Apr 01 2025

A382569 Decimal expansion of Sum_{p prime} 1/(p^3*(p + 1)^2).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.016480273853328955503361977914094146924354041689814015912823807...

Index to constants which are prime zeta sums {3,0,2}.

Cf. A085541, A085548, A179119, A382554.

sumeulerrat(1/(p^3*(p+1)^2)) \\ Amiram Eldar, Apr 01 2025

Equals -2*A085548 + A085541 + 3*A179119 - A382554.

Equals Sum_{k>=5} (-1)^(k+1) * (k-4) * P(k), where P is the prime zeta function. - Amiram Eldar, Apr 01 2025

A382571 Decimal expansion of Sum_{p prime} 1/((p - 1)*(p + 1)^2).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.153607996750202938743891326311996305247063362799933226638460591787...

Index to constants which are prime zeta sums {0,1,2}.

Cf. A136141, A179119, A382554.

sumeulerrat(1/((p-1)*(p+1)^2)) \\ Amiram Eldar, Apr 02 2025

Equals A136141/4 + A179119/4 - A382554/2.

Equals Sum_{k>=2} (k-1) * (P(2*k-1) - P(2*k)), where P is the prime zeta function. - Amiram Eldar, Apr 02 2025

A382573 Decimal expansion of Sum_{p prime} 1/((p - 1)^3*(p + 1)^2).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.119441349354286720549913256510757646460932...

Index to constants which are prime zeta sums {0,3,2}.

Cf. A086242, A136141, A179119, A380840, A382554.

sumeulerrat(1/((p-1)^3*(p+1)^2)) \\ Amiram Eldar, Apr 02 2025

Equals 3*A136141/16 - A086242/4 + A380840/4 + 3*A179119/16 - A382554/8.

Equals Sum_{k>=3} ((k-2)*(k-1)/2) * (P(2*k-1) + P(2*k)), where P is the prime zeta function. - Amiram Eldar, Apr 02 2025

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A086242 Decimal expansion of the sum of 1/(p-1)^2 over all primes p.

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A382555 Decimal expansion of Sum_{p prime} 1/(p + 1)^3.

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A324833 Decimal expansion of eta_2, a constant related to the asymptotic density of certain sets of residues.

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A324834 Decimal expansion of eta_3, a constant related to the asymptotic density of certain sets of residues.

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A110833 a(n) = (prime(n)+1)^2.

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A382567 Decimal expansion of Sum_{p prime} 1/(p*(p + 1)^2).

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A382568 Decimal expansion of Sum_{p prime} 1/(p^2*(p + 1)^2).

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A382569 Decimal expansion of Sum_{p prime} 1/(p^3*(p + 1)^2).

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