A380840 -id:A380840 - cp's OEIS frontend

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

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

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

A380689 Decimal expansion of Sum_{p prime} (p + 1)^3/((p - 1)^3*p^2).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Mar 31 2025

			7.86666539733180449247691932247069085597893471675885...

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

Cf. A085548, A086242, A136141, A380840.

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

Equals -A085548 + 6*A136141 - 4*A086242 + 8*A380840.

Equals P(2) + Sum_{k>=3} (4*k^2 - 16*k + 18) * P(k), where P is the prime zeta function. - Amiram Eldar, Apr 02 2025

A382550 Decimal expansion of Sum_{p prime} (p + 1)/(p - 1)^3.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Mar 31 2025

			3.67012319026579538178382074308670050246912943122554196...

Index to constants which are prime zeta sums {0,3,-1}.

Cf. A086242, A380840.

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

Equals A086242 + 2*A380840.

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

A382558 Decimal expansion of Sum_{p prime} p/(p - 1)^3.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Mar 31 2025

			2.522594092507215334850536936804570100824364695700429713...

Index to constants which are prime zeta sums {-1,3,0}.

Cf. A086242, A380840.

sumeulerrat(p/(p-1)^3) \\ Amiram Eldar, Apr 01 2025

Equals A086242 + A380840.

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

A382560 Decimal expansion of Sum_{p prime} 1/((p - 1)^3*p).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.545620772059739886880398135615633098083906111...

Index to constants which are prime zeta sums {1,3,0}.

Cf. A086242, A136141, A380840.

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

Equals A136141 - A086242 + A380840.

Equals Sum_{k>=4} ((k-3)*(k-2)/2) * P(k), where P is the prime zeta function. - Amiram Eldar, Apr 01 2025

A382561 Decimal expansion of Sum_{p prime} 1/((p - 1)^3*p^3).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.12976922838880491842498576468801517432330671985244961...

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

Cf. A085541, A085548, A086242, A136141, A380840.

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

Equals -3*A085548 - A085541 + 6*A136141 - 3*A086242 + A380840.

Equals Sum_{k>=6} ((k-5)*(k-4)/2) * P(k), where P is the prime zeta function. - Amiram Eldar, Apr 01 2025

A382566 Decimal expansion of Sum_{p prime} 1/((p - 1)^3*(p + 1)).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.36792162460638099759726137650329117056035747216338382601826906151734926635240156964828...

Index to constants which are prime zeta sums {0,3,1}.

Cf. A086242, A136141, A179119, A380840.

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

Equals A136141/8 - A086242/4 + A380840/2 + A179119/8.

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

A382572 Decimal expansion of Sum_{p prime} 1/((p - 1)^3*p^2).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.2646216953696293561853365599728302559685574801841430610540447238870242222757242396652...

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

Cf. A136141, A085548, A086242, A380840.

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

Equals -A085548 + 3*A136141 - 2*A086242 + A380840.

Equals Sum_{k>=5} ((k-4)*(k-3)/2) * P(k), where P is the prime zeta function. - Amiram Eldar, Apr 01 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

cp's OEIS Frontend

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

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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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A380689 Decimal expansion of Sum_{p prime} (p + 1)^3/((p - 1)^3*p^2).

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A382550 Decimal expansion of Sum_{p prime} (p + 1)/(p - 1)^3.

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A382558 Decimal expansion of Sum_{p prime} p/(p - 1)^3.

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A382560 Decimal expansion of Sum_{p prime} 1/((p - 1)^3*p).

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A382561 Decimal expansion of Sum_{p prime} 1/((p - 1)^3*p^3).

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A382566 Decimal expansion of Sum_{p prime} 1/((p - 1)^3*(p + 1)).

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