A090986 -id:A090986 - cp's OEIS frontend

A156648 Decimal expansion of Product_{k>=1} (1 + 1/k^2).

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

Offset: 1

R. J. Mathar, Feb 12 2009

Consider the value at s = 2 of the partition zeta functions zeta_{type}(s), where the defining sum runs over partitions into 'type' parts, where 'type' is 'even', 'prime' or 'distinct'. (For the precise definitions see R. Schneider's dissertation.) Then
  zeta_{even}(2) = Pi/2 = A019669;
  zeta_{prime}(2) = Pi^2/6 = A013661;
  zeta_{distinct}(2) = sinh(Pi)/Pi, this constant. - Peter Luschny, Aug 11 2021
For m>0, Product_{k>=1} (1 + m/k^2) = sinh(Pi*sqrt(m)) / (Pi*sqrt(m)). - Vaclav Kotesovec, Aug 30 2024

			3.676077910374977720695697492028260666507156346827630277478003593557447324111... = (1+1)*(1+1/4)*(1+1/9)*(1+1/16)*(1+1/25)*...

Reinhold Remmert, Classical topics in complex function theory, Vol. 172 of Graduate Texts in Mathematics, p. 12, Springer, 1997.

Robert Schneider, Eulerian series, zeta functions and the arithmetic of partitions, arXiv:2008.04243 [math.NT], 2020.

Square root of A084243.

Cf. A084243, A073017, A258870, A258871, A334411, A019669, A013661.

Maple
```
evalf(sinh(Pi)/Pi) ;
```

RealDigits[Sinh[Pi]/Pi, 10, 111][[1]] (* or *)
RealDigits[Re[1/(I!*(-I)!)], 10, 111][[1]] (* Robert G. Wilson v, Feb 23 2015 *)

sinh(Pi)/Pi \\ Charles R Greathouse IV, Dec 16 2013

prodnumrat(1 + 1/x^2, 1) \\ Charles R Greathouse IV, Feb 03 2025

Equals sinh(Pi)/Pi.
Equals 1/A090986. - R. J. Mathar, Mar 05 2009
Binomial(2, 1+i) = 1/(i!*(-i)!) (where x! means Gamma(x+1)). - Robert G. Wilson v, Feb 23 2015
Equals exp(Sum_{j>=1} (-(-1)^j*Zeta(2*j)/j)). - Vaclav Kotesovec, Mar 28 2019
Equals Product_{k>=1} (1+2/(k*(k+2))). - Amiram Eldar, Aug 16 2020

A212879 Decimal expansion of the absolute value of i!.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 29 2012

Also absolute value of Gamma(i).

			0.5215640468649398411581802696...

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A090986, A212877 (real(i!)), A212878 (-imag(i!)), A212880 (-arg(i!)).

N[Sqrt[Pi/Sinh[Pi]], 103] (* Vaclav Kotesovec, Dec 10 2015 *)
RealDigits[Abs[I!], 10, 120][[1]] (* Amiram Eldar, Oct 25 2024 *)

PARI
```
abs(gamma(I))
```

Equals abs(Gamma(i)), since i! = Gamma(1+i) = i*Gamma(i).
Equals sqrt(Pi/sinh(Pi)). - Vaclav Kotesovec, Dec 10 2015
Equals Product_{k>=1} k/sqrt(k^2+1) = sqrt(A090986). - Amiram Eldar, Oct 25 2024

A212877 Decimal expansion of the real part of i!, where i = sqrt(-1).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 29 2012

Also the negated imaginary part of Gamma(i).

			0.498015668118356042713691117462198...

Stanislav Sykora, Mathematical Constants, Stan's Library, Vol. II.
Wikipedia, Particular values of the Gamma function

Cf. A212878 (-imag(i!)), A212879 (abs(i!)), A212880 (-arg(i!)), A090986.

RealDigits[Re[Gamma[I + 1]], 10, 105] (* T. D. Noe, May 29 2012 *)

PARI
```
real(I*gamma(I))
```

i! = gamma(1+i) = i*gamma(i).
Equals (1/2)*Integral_{x=-1/e..0} LambertW(x)*sin(log(-LambertW(x)))-LambertW(-1,x)*sin(log(-LambertW(-1,x))) dx. - Gleb Koloskov, Oct 01 2021
Equals Integral_{x=0..+oo} exp(-x)*cos(log(x)) dx. - Jianing Song, Sep 27 2023
A212877^2 + A212878^2 = A090986 = Pi/sinh(Pi). - Vaclav Kotesovec, Dec 28 2023

A212878 Decimal expansion of the negated imaginary part of i!.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 29 2012

Also the negated real part of Gamma(i).

			0.15494982830181068512495513048...

Cf. A212877 (real(i!)), A212879 (abs(i!)), A212880 (-arg(i!)), A090986.

-Gamma[I] // Re // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, May 13 2013 *)

PARI
```
-imag(I*gamma(I))
```

i! = gamma(1+i) = i*gamma(i).
Equals -Integral_{x=0..+oo} exp(-x)*sin(log(x)) dx. - Jianing Song, Sep 27 2023
A212877^2 + A212878^2 = A090986 = Pi/sinh(Pi). - Vaclav Kotesovec, Dec 28 2023

A269791 G.f.: Product_{n>=1} 1/(1 - x^n/n^4) = Sum_{n>=0} a(n)*x^n/n!^4.

Original entry on oeis.org

1, 1, 17, 1393, 359200, 224991776, 291968881696, 701412781560352, 2873957814268080128, 18859650596161401139200, 188619789441121624152354816, 2761804817165898231731040301056, 57271995555712767650976765232545792, 1635810412682066454426684822491878391808

Offset: 0

Vaclav Kotesovec, Mar 05 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..140

Cf. A007841, A249588, A249593, A269793, A269794.

Table[n!^4 * SeriesCoefficient[Product[1/(1 - x^k/k^4), {k, 1, n}], {x, 0, n}], {n, 0, 20}]

{a(n)=n!^4*polcoeff(prod(k=1, n, 1/(1-x^k/k^4 +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

a(n) ~ c * n!^4, where c = Product_{k>=2} 1/(1-1/k^4) = 4*Pi/sinh(Pi) = 4*A090986 = 1.08811621992853265180094633468815...

A090087 Smallest odd pseudoprimes to base n, not necessarily exceeding n. Compare with A007535 and A090086.

Original entry on oeis.org

9, 341, 91, 15, 217, 35, 25, 9, 91, 9, 15, 65, 21, 15, 341, 15, 9, 25, 9, 21, 55, 21, 33, 25, 39, 9, 65, 9, 15, 49, 15, 25, 85, 15, 9, 35, 9, 39, 95, 39, 15, 205, 21, 9, 133, 9, 65, 49, 15, 21, 25, 51, 9, 55, 9, 15, 25, 57, 15, 341, 15, 9, 341, 9, 33, 65, 33, 25, 35, 69, 9, 85, 9, 15

Offset: 1

Labos Elemer, Nov 25 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007535, A090986, A090088, A090089.

a[n_] := Module[{k = 1}, While[GCD[n, k] > 1 || PrimeQ[k] || PowerMod[n, k - 1, k] != 1, k += 2]; k]; Array[a, 100] (* Amiram Eldar, Nov 11 2019 *)

a(n)=Min{x=odd number; Mod[ -1+n^(x-1), x]=0}

A364355 Decimal expansion of the unique value of x such that Gamma(-x + i*sqrt(1-x^2)) is a real number and -1 < x < 1.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Jul 20 2023

Gamma(-A364355 + i*sqrt(1-A364355^2)) = -0.6749332470449905963531... see A364356.

Also decimal expansion of the unique value of x in the range -1 < x < 1 for which the function Re(Gamma(-x + i*sqrt(1-x^2)))/abs(Gamma(-x + i*sqrt(1-x^2))) is minimized.

			x = 0.54197987169489060244332278779...

Cf. A090986, A212877, A212878, A212880, A212879, A364356.

RealDigits[x /. FindRoot[Im[Gamma[-x + I Sqrt[1 - x^2]]], {x, 0.5}, WorkingPrecision -> 106]][[1]]

A090088 Smallest even pseudoprimes to odd base=2n-1, not necessarily exceeding n. See also A007535 and A090086, A090087.

Original entry on oeis.org

4, 286, 4, 6, 4, 10, 4, 14, 4, 6, 4, 22, 4, 26, 4, 6, 4, 34, 4, 38, 4, 6, 4, 46, 4, 10, 4, 6, 4, 58, 4, 62, 4, 6, 4, 10, 4, 74, 4, 6, 4, 82, 4, 86, 4, 6, 4, 94, 4, 14, 4, 6, 4, 106, 4, 10, 4, 6, 4, 118, 4, 122, 4, 6, 4, 10, 4, 134, 4, 6, 4, 142, 4, 146, 4, 6, 4, 14, 4, 158, 4, 6, 4, 166, 4, 10

Offset: 1

Labos Elemer, Nov 25 2003

nonn

For an even base there are no even pseudoprimes.

			n=2, 2n-2=3 as base, smallest relevant power is -1+2^(286-1) which is divisible by 286.

Antti Karttunen, Table of n, a(n) for n = 1..16520
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A007535, A090986, A090987, A090089.

Array[Block[{k = 4}, While[PowerMod[2 # - 1, k - 1, k] != 1, k += 2]; k] &, 86] (* Michael De Vlieger, Nov 13 2018 *)

A090088(n) = { forstep(k=4, oo, 2, if(1==(Mod(n+n-1, k)^(k-1)), return (k)); ); } \\ (After code in A090086) - Antti Karttunen, Nov 10 2018

a(n) = Min_{x=even number; (-1 + n^(x-1)) mod x = 0}.

A090089 Smallest even pseudoprimes to odd base=4n-1, not necessarily exceeding n.

Original entry on oeis.org

286, 6, 10, 14, 6, 22, 26, 6, 34, 38, 6, 46, 10, 6, 58, 62, 6, 10, 74, 6, 82, 86, 6, 94, 14, 6, 106, 10, 6, 118, 122, 6, 10, 134, 6, 142, 146, 6, 14, 158, 6, 166, 10, 6, 178, 14, 6, 10, 194, 6, 202, 206, 6, 214, 218, 6, 226, 10, 6, 14, 22, 6, 10, 254, 6, 262, 14, 6, 274, 278, 6

Offset: 1

Labos Elemer, Nov 25 2003

nonn

There are no even pseudoprimes to an even base.

			n=1: base = 4n-1=3, smallest relevant power is -1+2^(286-1) which is divisible by 286.
Sieving further residue classes, smallest regularly arising pseudoprimes are 6,10 etc..

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007535, A090085, A090986, A090087, A090088.

a[n_] := Module[{k = 2}, While[GCD[n, k] > 1 || PrimeQ[k] || PowerMod[n, k - 1, k] != 1, k += 2]; k]; Table[a[4*n - 1], {n, 1, 100}] (* Amiram Eldar, Nov 11 2019 *)

a(n)=Min{x=4n-1 number; Mod[ -1+n^(x-1), x]=0}

A112407 Decimal expansion of a semiprime analog of a Ramanujan formula.

Original entry on oeis.org

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

Offset: 0

Jonathan Vos Post, Dec 21 2005

This is related to Ramanujan's surprising formula: Prod[from n = 1 to infinity] (prime(n)^2 - 1)/ (prime(n)^2 + 1) = 2/5 and we use it in finding A112407 as the semiprime analog. We also use: A090986 = Decimal expansion of Pi csch Pi = Prod[from n = 2 to infinity] (n^2 - 1)/(n^2 + 1).
Since every integer above 1 is a k-almost prime for some k, we factor the (n^2 - 1)/(n^2 + 1) infinite product and use Ramanujan's formula, to have: Prod[from n = 1 to infinity] (prime(n)^2-1)/(prime(n)^2+1) * Prod[from n = 1 to infinity] (semiprime(n)^2 - 1)/(semiprime(n)^2 + 1) * Prod[from n = 1 to infinity] (3-almostprime(n)^2 - 1)/ (3-almostprime(n)^2 + 1) * ... * Prod[from n = 1 to infinity] (k-almostprime(n)^2 - 1)/ (k-almostprime(n)^2 + 1) * ... = Prod[from n = 2 to infinity] (n^2 - 1)/(n^2 + 1) = pi csch pi as each integer appear once and only once in numerator and once and only once in denominator.
2/5 is the first (Ramanujan, prime) term in this infinite product of infinite products. This here is the second (semiprime) term. A155799 is the third (3-almost prime) term. All of these have slow convergence.

			0.75449970170951407835571816895054...

Borwein, J.; Bailey, D.; and Girgensohn, R. "Two Products." Section 1.2 in Experimentation in Mathematics: Computational Paths to Discovery. Natick, MA: A. K. Peters, pp. 4-7, 2004.

R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514, Table 1, k=s=2.
Eric Weisstein's World of Mathematics, Infinite Product.
Eric Weisstein's World of Mathematics, Hyperbolic Cosecant

Cf. A001358, A090986, A114529-A114536.

spQ[n_] := Plus @@ Last /@ FactorInteger@n == 2; p = 1; Do[If[spQ[n], p = N[p*(n^2 - 1)/(n^2 + 1), 64]], {n, 10^6}]; p (* Robert G. Wilson v *) (* This program converges slowly. For {n, 10^6}, only the first 6 digits are correct. - Jason Yuen, Aug 10 2025 *)

A(lim)=my(x=1.);forprime(p=2,lim\2,forprime(q=2,min(p,lim\p),x*=1-2/((p*q)^2+1)));x \\ Charles R Greathouse IV, Aug 15 2011

Decimal expansion of a = prod[from n = 1 to infinity] (semiprime(n)^2 - 1)/(semiprime(n)^2 + 1) = prod[from n = 1 to infinity] (A001358(n)^2 - 1)/(A001358(n)^2 + 1).

log a = -2*sum_{l=1..infinity} P_2(2*(2l-1))/(2l-1), where P_k(s) are the k-almost prime zeta functions of arXiv:0803.0900 [math.NT]. - R. J. Mathar, Jan 27 2009

Edited and extended by R. J. Mathar, Jan 27 2009

cp's OEIS Frontend

A156648 Decimal expansion of Product_{k>=1} (1 + 1/k^2).

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A212879 Decimal expansion of the absolute value of i!.

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A212877 Decimal expansion of the real part of i!, where i = sqrt(-1).

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A212878 Decimal expansion of the negated imaginary part of i!.

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A269791 G.f.: Product_{n>=1} 1/(1 - x^n/n^4) = Sum_{n>=0} a(n)*x^n/n!^4.

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A090087 Smallest odd pseudoprimes to base n, not necessarily exceeding n. Compare with A007535 and A090086.

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A364355 Decimal expansion of the unique value of x such that Gamma(-x + i*sqrt(1-x^2)) is a real number and -1 < x < 1.

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A090088 Smallest even pseudoprimes to odd base=2n-1, not necessarily exceeding n. See also A007535 and A090086, A090087.