A111003 -id:A111003 - cp's OEIS frontend

A340065 Decimal expansion of the Product_{p>=2} 1+p^2/((p-1)^2*(p+1)^2) where p are successive prime numbers A000040.

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

Offset: 1

Artur Jasinski, Dec 28 2020

This is a rational number.
This constant does not belong to the infinite series of prime number products of the form: Product_{p>=2} (p^(2*n)-1)/(p^(2*n)+1),
which are rational numbers equal to zeta(4*n)/(zeta(2*n))^2 = A114362(n+1)/A114363(n+1).
This number has decimal period length 230:
1.81(0781476121562952243125904486251808972503617945007235890014471780028943
5600578871201157742402315484804630969609261939218523878437047756874095
5137481910274963820549927641099855282199710564399421128798842257597684
51519536903039073806).

			1.8107814761215629522431259...

Cf. A000040, A005361, A084920, A065483, A065484, A065485, A109695, A111003, A114362, A114363, A116393, A167864, A231535, A307868, A330523, A330595, A335319, A335762, A335818, A339925, A340066.

Mathematica
```
RealDigits[N[5005/2764,105]][[1]]
```

default(realprecision,105)
prodeulerrat(1+p^2/((p-1)^2*(p+1)^2))

Equals 5005/2764 = 5*7*11*13/(2^2*691).
Equals Product_{n>=1} 1+A000040(n)^2/A084920(n)^2.
Equals (13/9)*A340066.
From Vaclav Kotesovec, Dec 29 2020: (Start)
Equals 3/2 * (Product_{p prime} (p^6+1)/(p^6-1)) * (Product_{p prime} (p^4+1)/(p^4-1)).
Equals 7*zeta(6)^2 / (4*zeta(12)).
Equals -7*binomial(12, 6) * Bernoulli(6)^2 / (8*Bernoulli(12)). (End)
Equals Sum_{k>=1} A005361(k)/k^2. - Amiram Eldar, Jan 23 2024

A362984 Decimal expansion of the asymptotic mean of the abundancy index of the powerful numbers (A001694).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 12 2023

The abundancy index of a positive integer k is A000203(k)/k = A017665(k)/A017666(k).

The asymptotic mean of the abundancy index over all the positive integers is lim_{m->oo} (1/m) * Sum_{k=1..m} A000203(k)/k = Pi^2/6 = zeta(2) = 1.644934... (A013661).

			2.14968690301526765128219042105109416145987653275100999873...

Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (6).

Cf. A000203, A001694, A017665, A017666, A180114.

Similar constants (the asymptotic mean of the abundancy index of other sequences): A013661 (all positive integers), A082020 (cubefree), A111003 (odd), A157292 (5-free), A157294 (7-free), A157296 (9-free), A240976 (squares), A245058 (even), A306633 (squarefree), A362985 (cubefull).

$MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{2, -3, 4, -6, 7, -7, 7, -6, 5, -3, 2, -1}, {0, 0, 0, 4, 5, 6, 0, -12, -9, -5, 0, 22}, m]; RealDigits[(2^4 + 2^2 + 2^(3/2) - 1)/(2^4 - 2)*(3^4 + 3^2 + 3^(3/2) - 1)/(3^4 - 3) * Exp[NSum[Indexed[c, n]*(PrimeZetaP[n/2] - 1/2^(n/2) - 1/3^(n/2))/n, {n, 4, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 120][[1]]

prodeulerrat((p^8 + p^4 + p^3 - 1)/(p^8 - p^2), 1/2)

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A180114(k)/A001694(k).

Equals Product_{p prime} (p^4 + p^2 + p^(3/2) - 1)/(p^4 - p) = Product_{p prime} (1 + (p^2 + p^(3/2) + p - 1)/(p^4 - p)) (Jakimczuk and Lalín, 2022).

A362985 Decimal expansion of the asymptotic mean of the abundancy index of the cubefull numbers (A036966).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 12 2023

			2.48217919642235952546167643674687698536368940971930468354...

Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (6).

Cf. A000203, A036966, A017665, A017666, A362986.

Similar constants (the asymptotic mean of the abundancy index of other sequences): A013661 (all positive integers), A082020 (cubefree), A111003 (odd), A157292 (5-free), A157294 (7-free), A157296 (9-free), A245058 (even), A240976 (squares), A306633 (squarefree), A362984 (powerful).

$MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{2, -1, -2, 3, -2, -1, 3, -2, -2, 3, -1, -2, 3, -1, -1, 1}, {0, 0, 0, -4, 0, 6, 7, 4, 9, 0, -11, -22, -26, -21, -15, 20}, m]; RealDigits[((2^5 + 2^(10/3) + 2^3 + 2^(8/3) - 1)/(2^(10/3)*(2^(5/3) + 2^(1/3) + 1)))*((3^5 + 3^(10/3) + 3^3 + 3^(8/3) - 1)/(3^(10/3)*(3^(5/3) + 3^(1/3) + 1))) * Zeta[4/3] * Exp[NSum[Indexed[c, n]*(PrimeZetaP[n/3] - 1/2^(n/3) - 1/3^(n/3))/n, {n, 4, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 120][[1]]

zeta(4/3) * prodeulerrat((p^15 + p^10 + p^9 + p^8 - 1)/(p^10 * (p^5 + p + 1)), 1/3)

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A362986(k)/A036966(k).

Equals zeta(4/3) * Product_{p prime} ((p^5 + p^(10/3) + p^3 + p^(8/3) - 1)/(p^(10/3) * (p^(5/3) + p^(1/3) + 1))).

A282468 Decimal expansion of the zeta function at 2 of every second prime number.

Original entry on oeis.org

1, 4, 4, 7, 1, 5, 5, 8, 6, 6, 8, 8, 7

Offset: 0

Terry D. Grant, Apr 14 2017

From Husnain Raza, Aug 30 2023: (Start)
Note that since p_n > n*log(n), we can place a bound on the tail of the sum:
Sum_{n >= N} (prime(2n))^(-2) <= Sum_{n >= N} (2*n*log(2n))^(-2) <= Integral_{x=N..oo} (2*x*log(2x))^(-2) dx.
Taking the sum over all primes < 10^12, we see that the constant lies between 0.14471558668870 and 0.14471558668873. (End)

			1/3^2 + 1/7^2 + 1/13^2 + 1/19^2 + 1/29^2 + ... = 0.14471558...

Husnain Raza, PARI program for calculating more digits.

Cf. A031215, A085548.

Zeta functions at 2: A085548 (for primes), A275647 (for nonprimes), A013661 (for natural numbers), A117543 (for semiprimes), A131653 (for triprimes), A222171 (for even numbers), A111003 (for odd numbers).

PARI
```
sum(n=1, 2500000, 1./prime(2*n)^2)
```
PARI
```
\\ see Raza link
```

Equals Sum_{n>=1} 1/A031215(n)^2 = Sum_{n>=1} 1/prime(2n)^2.

a(8)-a(12) from Husnain Raza, Aug 31 2023

A331095 Decimal expansion of 32/Pi^3.

Original entry on oeis.org

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

Offset: 1

Dimitris Valianatos, Jan 08 2020

For odd prime numbers: Product_{odd primes p} 1/(1 - 1/p^2) = Pi^2/8 = (3/4)*zeta(2) = A111003.

For odd composite numbers: Product_{odd composite numbers c} 1/(1 - 1/c^2) = (81/80) * (225/224) * (441/440) * (625/624) * (729/728) * ... = 32/Pi^3, this constant.

			1.032049101862383653901505686034038...

Index entries for transcendental numbers

Cf. A088538 (4/Pi), A111003 (Pi^2/8).

RealDigits[32/Pi^3, 10, 100][[1]] (* Amiram Eldar, Jan 10 2020 *)

p = 1.0; forstep(n = 3, 10^7, 2, if(!isprime(n), p*= (1 / (1 - 1 / n^2)))); print(p)

PARI
```
32/Pi^3
```

Equals A088538/A111003.

A340066 Decimal expansion of the Product_{p>=3} 1+p^2/((p-1)^2*(p+1)^2) where p are successive prime numbers A000040.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Dec 28 2020

This is a rational number.
This constant does not belong to the infinite series of prime number products of the form: Product_{p>=2} (p^(2*n)-1)/(p^(2*n)+1),
which are rational numbers equal to zeta(4*n)/zeta^2(2*n) = A114362(n+1)/A114363(n+1).
This number has decimal period length 230:
1.25(3617945007235890014471780028943560057887120115774240231548480463096960
9261939218523878437047756874095513748191027496382054992764109985528219
9710564399421128798842257597684515195369030390738060781476121562952243
12590448625180897250).

			1.25361794500723589001447178...

Cf. A065483, A065484, A065485, A109695, A111003, A114362, A114363, A116393, A167864, A231535, A307868, A330523, A330595, A335319, A335762, A335818, A339925, A340065.

Mathematica
```
RealDigits[N[3465/2764, 105]][[1]]
```

default(realprecision, 105)
prodeulerrat(1+p^2/((p-1)^2*(p+1)^2),1,3)

Equals 3465/2764 = 3^2*5*7*11/(2^2*691).
Equals Product_{n>=2} 1+A000040(n)^2/A084920(n)^2.
Equals (9/13)*A340065.

A345204 Decimal expansion of 5/4 + Pi^2/8 + zeta(3).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 10 2021

			3.68575745329576411275404953649596888267919871824559...

Ovidiu Furdui, Series Involving Products of Two Harmonic Numbers, Mathematics Magazine, Vol. 84, No. 5 (2011), pp. 371-377.

Cf. A001008, A002117, A002805, A111003, A218505, A345203.

RealDigits[5/4 + Pi^2/8 + Zeta[3], 10, 100][[1]]

5/4 + Pi^2/8 + zeta(3) \\ Stefano Spezia, Jan 30 2025

Equals Sum_{k>=1} H(k)*H(k+2)/(k*(k+2)), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number (Furdui, 2011).

Equals 5/4 + A111003 + A002117.

A340565 Decimal expansion of the Product_{lesser twin primes p == 5 (mod 6)} 1/(1 - 1/p^2).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jan 11 2021

Lesser twin primes A001359 (with the exception of the first prime, 3) are congruent to 5 mod 6: this constant is smaller than A340576.
By extrapolating method most probably the next two decimal digits are 1.056932291(46).
The known high-precision algorithms for Euler products are based on the Dirichlet L function and the Moebius inversion formula (see Mathematica procedure of Jean-François Alcover in A175646).
The constant is between 1.056932291453... and 1.056932291494. - R. J. Mathar, Feb 14 2025

			1.0569322914...

Cf. A005596, A065463, A065465, A065483, A065485, A065489, A065493, A072691, A082695, A111003, A175646.

One more digit confirmed by a bracketing of partial products - R. J. Mathar, Feb 14 2025

A348731 Decimal expansion of Integral_{x=0..1} x*log(x)/(1+x+x^2) dx (negated).

Original entry on oeis.org

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

Offset: 0

Dumitru Damian, Oct 31 2021

			-0.15766014916783233039054467406996221822374946546295676913413604497322566...

Cf. A072691, A086722, A086724, A111003, A222171.

RealDigits[Integrate[x*Log[x]/(1 + x + x^2), {x, 0, 1}], 10, 100][[1]] (* Amiram Eldar, Oct 31 2021 *)
RealDigits[Pi^2/54 - PolyGamma[1, 2/3]/9, 10, 100][[1]] (* Vaclav Kotesovec, Oct 31 2021 *)

intnum(x=0, 1, x*log(x)/(1+x+x^2)) \\ Michel Marcus, Oct 31 2021

RealField(25)(numerical_integral(x*log(x)/(1+x+x^2), 0, 1)[0])

Equals Pi^2/54 - PolyGamma(1, 2/3)/9. - Vaclav Kotesovec, Oct 31 2021

A348763 Decimal expansion of Sum_{n>=1} ((-1)^(n+1)*n)/(n+1)^2.

Original entry on oeis.org

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

Offset: 0

Dumitru Damian, Oct 31 2021

			0.12931985286416790881897546186483602653397481624314396474709910519161011...

Eric Weisstein's World of Mathematics, Dilogarithm
Index entries for zeta function.

Cf. A006752, A072691, A111003, A153071, A175571, A181983, A300707.

RealDigits[Pi^2/12 - Log[2], 10, 100][[1]] (* Amiram Eldar, Nov 30 2021 *)

-sumalt(n=1, (-1)^n*n/(n+1)^2) \\ Charles R Greathouse IV, Nov 01 2021

Pi^2/12-log(2) \\ Charles R Greathouse IV, Nov 01 2021

from scipy.special import zeta
from math import log
int(''.join(n for n in list(str(zeta(2)/2-log(2)))[2:-2]))

int(str(sum((-1)**(n+1)*n/(n+1)**2 for n in range(1,5000000)))[2:-2])

SageMath
```
(pi^2/12-log(2)).n(digits=100)
```

Equals Pi^2/12-log(2).
Equals Sum_{k>=2} (zeta(k)-zeta(k+1))/2^k. - Amiram Eldar, Mar 20 2022
Equals Integral_{x >= 0} x/(1 + exp(x))^2 dx = (1/2) * Integral_{x >= 0} x*(x - 2)*exp(x)/(1 + exp(x))^2 dx . - Peter Bala, Apr 26 2025

cp's OEIS Frontend

A340065 Decimal expansion of the Product_{p>=2} 1+p^2/((p-1)^2*(p+1)^2) where p are successive prime numbers A000040.

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A362984 Decimal expansion of the asymptotic mean of the abundancy index of the powerful numbers (A001694).

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A362985 Decimal expansion of the asymptotic mean of the abundancy index of the cubefull numbers (A036966).

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A282468 Decimal expansion of the zeta function at 2 of every second prime number.

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A331095 Decimal expansion of 32/Pi^3.

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A340066 Decimal expansion of the Product_{p>=3} 1+p^2/((p-1)^2*(p+1)^2) where p are successive prime numbers A000040.

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A345204 Decimal expansion of 5/4 + Pi^2/8 + zeta(3).

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