A382555 -id:A382555 - cp's OEIS frontend

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

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

A382575 Decimal expansion of Sum_{p prime} (p - 1)/(p + 1)^3 .

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.12304471118232056125715117369116831819672016343537931371506541387358...

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

Cf. A382554, A382555.

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

Equals A382554 - 2*A382555.

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

A382576 Decimal expansion of Sum_{p prime} p/(p + 1)^3.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.183761007669456934104549772519259642917046907807070556334380782178702616004...

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

Cf. A382554, A382555.

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

Equals A382554 - A382555.

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

A382577 Decimal expansion of Sum_{p prime} 1/(p*(p + 1)^3).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.025036325620473561215747617911302468286559022923617414577217081973488...

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

Cf. A179119, A382554, A382555.

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

Equals A179119 - A382554 - A382555.

Equals Sum_{k>=4} (-1)^k * ((k-3)(k-2)/2) * P(k), where P is the prime zeta function. - Amiram Eldar, Apr 02 2025

A382578 Decimal expansion of Sum_{p prime} 1/(p^2*(p + 1)^3).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.01122854604877876221255494209480691223552713354689...

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

Cf. A085548, A179119, A382554, A382555.

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

Equals A085548 - 3*A179119 + 2*A382554 + A382555.

Equals Sum_{k>=5} (-1)^(k+1) * ((k-4)(k-3)/2) * P(k), where P is the prime zeta function. - Amiram Eldar, Apr 02 2025

A382579 Decimal expansion of Sum_{p prime} 1/(p^3*(p + 1)^3).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.00525172780455019329080703581928723468882690814291812210038301...

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

Cf. A085548, A085541, A179119, A382554, A382555.

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

Equals -3*A085548 + A085541 + 6*A179119 - 3*A382554 - A382555.

Equals Sum_{k>=6} (-1)^k * ((k-5)(k-4)/2) * P(k), where P is the prime zeta function. - Amiram Eldar, Apr 01 2025

A382580 Decimal expansion of Sum_{p prime} 1/((p - 1)*(p + 1)^3).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.046445850131533282948246363741952490263368309214120992009572611741379098614202478270928...

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

Cf. A136141, A179119, A382554, A382555.

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

Equals A136141/8 + A179119/8 - A382554/4 - A382555/2.

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

A382581 Decimal expansion of Sum_{p prime} 1/((p - 1)^2*(p + 1)^3).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.0412965378831371367745942498699116936875624524925423481454200464113007835583572808145218...

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

Cf. A086242, A136141, A179119, A382554, A382555.

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

Equals -3*A136141/16 + A086242/8 - 3*A179119/16 + A382554/4 + A382555/4.

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

A382601 Decimal expansion of Sum_{p prime} 1/((p - 1)p(p + 1)^3).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.02140952451105972173249874583065002197680928629050357743...

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

Cf. A136141, A179119, A382554, A382555.

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

Equals A136141/8 - 7*A179119/8 + 3*A382554/4 + A382555/2.

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

A382602 Decimal expansion of Sum_{p prime} 1/((p - 1)p^2(p + 1)^3).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.010180978462280959519943803735843109741282152...

Index to constants which are prime zeta sums {2,1,3}.

Cf. A085548, A136141, A179119, A382554, A382555.

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

Equals -A085548 + A136141/8 + 17*A179119/8 - 5*A382554/4 - A382555/2.

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

cp's OEIS Frontend

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

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A382575 Decimal expansion of Sum_{p prime} (p - 1)/(p + 1)^3 .

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A382576 Decimal expansion of Sum_{p prime} p/(p + 1)^3.

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A382577 Decimal expansion of Sum_{p prime} 1/(p*(p + 1)^3).

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A382578 Decimal expansion of Sum_{p prime} 1/(p^2*(p + 1)^3).

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A382579 Decimal expansion of Sum_{p prime} 1/(p^3*(p + 1)^3).

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A382580 Decimal expansion of Sum_{p prime} 1/((p - 1)*(p + 1)^3).

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A382581 Decimal expansion of Sum_{p prime} 1/((p - 1)^2*(p + 1)^3).

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A382601 Decimal expansion of Sum_{p prime} 1/((p - 1)p(p + 1)^3).

A382602 Decimal expansion of Sum_{p prime} 1/((p - 1)p^2(p + 1)^3).