A128059 -id:A128059 - cp's OEIS frontend

A356360 Denominator of the continued fraction 1/(2-3/(3-4/(4-5/(...(n-1)-n/(n+1))))).

5, 7, 3, 11, 13, 1, 17, 19, 1, 23, 1, 1, 29, 31, 1, 1, 37, 1, 41, 43, 1, 47, 1, 1, 53, 1, 1, 59, 61, 1, 1, 67, 1, 71, 73, 1, 1, 79, 1, 83, 1, 1, 89, 1, 1, 1, 97, 1, 101, 103, 1, 107, 109, 1, 113, 1, 1, 1, 1, 1, 1, 127, 1, 131, 1, 1, 137, 139, 1, 1, 1, 1, 149, 151, 1, 1, 157, 1, 1, 163, 1, 167

Offset: 3

Mohammed Bouras, Oct 15 2022

nonn

Conjecture: The sequence contains only 1's and the primes.
Similar continued fraction to A356247.
Same as A128059(n), A145737(n-1) and A097302(n-2) for n > 5.
a(n) = 1 positions appear to correspond to A104275(m), m > 2. Conjecture: all odd primes are seen in order after 11. - Bill McEachen, Aug 05 2024

Mohammed Bouras, The Distribution Of Prime Numbers And The Continued Fractions, (2022).

Cf. A051403, A051417, A097302, A128059, A145737, A356247.

Cf. A104275.

from math import gcd, factorial
def A356360(n): return (a:=(n<<1)-1)//gcd(a, a*sum(factorial(k) for k in range(n-2))+n*factorial(n-2)>>1) # Chai Wah Wu, Feb 26 2024

For n >= 3, the formula of the continued fraction is as follows:
(A051403(n-2) + A051403(n-3))/(2n - 1) = 1/(2-3/(3-4/(4-5/(...(n-1)-n/(n+1))))).
a(n) = (2n - 1)/gcd(2n - 1, A051403(n-2) + A051403(n-3)).
From the conjecture: Except for n = 5, a(n)= 2n - 1 if 2n-1 is prime, 1 otherwise.

A128060 a(n) = 2n - numerator((2n-1)^2/(2(2n)!)).

Original entry on oeis.org

-1, 1, 1, 1, 1, 9, 1, 1, 15, 1, 1, 21, 1, 25, 27, 1, 1, 33, 35, 1, 39, 1, 1, 45, 1, 49, 51, 1, 55, 57, 1, 1, 63, 65, 1, 69, 1, 1, 75, 77, 1, 81, 1, 85, 87, 1, 91, 93, 95, 1, 99, 1, 1, 105, 1, 1, 111, 1, 115, 117, 119, 121, 123, 125, 1, 129, 1, 133, 135, 1, 1, 141, 143, 145, 147

Offset: 0

Paul Barry, Feb 13 2007

Odd composite numbers with placeholders for primes between them.

Ivan V. Morozov, On Quotients of a More General Theorem of Wilson, arXiv:2505.16201 [math.NT], 2025. See denominators of Z+ p. 9.

Cf. A128059, A160479, A217983.

A128060 := proc(n): 2*n - numer((2*n-1)^2/(2*(2*n)!)) end: seq(A128060(n), n=0..62);
# End program 1 [Johannes W. Meijer, Oct 25 2012]
A128060 := proc(n) local n1: n1:=2*n-1: if type(n1, prime) then A128060(n) := 1 else A128060(n) := n1 fi: end: seq(A128060(n), n=0..62);
# End program 2 [Johannes W. Meijer, Oct 25 2012]

Table[2n - Numerator[(2n - 1)^2/(2(2n)!)], {n, 0, 74}] (* Alonso del Arte, Jan 05 2014 *)

a(n) = 2*n - A128059(n).
a(n) = (A217983(n-1) * (2*n-1))/A160479(n) for n >= 3. - Johannes W. Meijer, Oct 25 2012
a(0) = -1, a(n) = gcd(2*n-1, (2*n-2)!), n > 0. - Wesley Ivan Hurt, Jan 05 2014

More terms from Michel Marcus, May 23 2025

A145737 a(n) = square part of A145609(n).

Original entry on oeis.org

1, 5, 7, 1, 11, 13, 1, 17, 19, 1, 23, 1, 1, 29, 31, 1, 1, 37, 1, 41, 43, 1, 47, 1, 1, 53, 1, 1, 59, 61, 1, 1, 67, 1, 71, 73, 1, 1, 79, 1, 83, 1, 1, 89, 1, 1, 1, 97, 1, 101, 103, 1, 107, 109, 1, 113, 1, 1, 1, 1, 1, 1, 127, 1, 131, 1, 1, 137, 139, 1, 1, 1, 1, 149, 151, 1, 1, 157, 1, 1, 163

Offset: 1

Artur Jasinski, Oct 17 2008

nonn

For squarefree parts see A145738. A128059 is a very similar sequence.

Cf. A008833, A128059, A145609, A145738.

seq(denom(n!*3*n*(n+1)/(2*(2*n+1))), n=1..81); # Gary Detlefs, Oct 18 2011

m = 1; aa = {}; Do[k = 0; Do[k = k + m^(2 r + 1 - d)/d, {d, 1, 2 r}]; b = Sqrt[Numerator[k]] /. Sqrt[] -> 1; AppendTo[aa, b], {r, 1, 137}]; aa (* _Artur Jasinski *)

from sympy import isprime
def A145737(n): return a if isprime(a:=(n<<1)+1) and n>1 else 1 # Chai Wah Wu, Feb 26 2024

a(n) = 2n+1 if 2n+1 is prime, 1 otherwise, for n > 1.
From Gary Detlefs, Oct 18 2011: (Start)
a(n) = Denominator(n!*(Sum_{k=1..n} k^3)/(Sum_{k=1..n} k^2))
     = Denominator(n!*3*n*(n+1)/(2*(2*n+1))). (End)

A145738 a(n) = squarefree part of A145609(n).

Original entry on oeis.org

3, 1, 1, 761, 61, 509, 1171733, 8431, 39541, 55835135, 36093, 1347822955, 34395742267, 375035183, 9682292227, 586061125622639, 54062195834749, 40030624861, 2053580969474233, 1236275063173, 6657281227331

Offset: 1

Artur Jasinski, Oct 17 2008

nonn

Cf. A128059, A145609, A145738, A145737 (for square parts).

m = 1; aa = {}; Do[k = 0; Do[k = k + m^(2 r + 1 - d)/d, {d, 1, 2 r}]; b = Sqrt[Numerator[k]] /. Sqrt[] -> 1; c = Sqrt[Numerator[k]]/b /. Sqrt[x] -> x; AppendTo[aa, c], {r, 1, 25}]; aa

A347425 a(n) = Bernoulli(2n) (2n+1)! if 2n+1 is a prime, otherwise a(n) = Bernoulli(2n) (2*n)!.

Original entry on oeis.org

1, 1, -4, 120, -1344, 3024000, -1576143360, 101708006400, -2522591034163200, 6686974460694528000, -1287307431968882688000, 160078872315904478576640000, -53718579665963356985229312000, 574898901006059006921736192000000, -241364461951740682229320388129587200000

Offset: 0

Ilya Gutkovskiy, Sep 01 2021

			Bernoulli(2*n) * (2*n)! = [ 1, 1/3, -4/5, 120/7, -1344, 3024000/11, -1576143360/13, 101708006400, -2522591034163200/17, 6686974460694528000/19, ... ].

Index entries for sequences related to Bernoulli numbers

Cf. A000367, A001332, A002431, A002445, A036278, A128059.

a[n_] := If[PrimeQ[2 n + 1], BernoulliB[2 n] (2 n + 1)!, BernoulliB[2 n] (2 n)!]; Table[a[n], {n, 0, 14}]
Table[Numerator[BernoulliB[2 n] (2 n)!], {n, 0, 14}]
Table[Numerator[(2 n)!^2 SeriesCoefficient[x Coth[x/2]/2, {x, 0, 2 n}]], {n, 0, 14}]
b[0] = 1; b[n_] := b[n] = -Sum[Binomial[n, k]^2 k! b[n - k]/(k + 1), {k, 1, n}]; a[n_] := Numerator[b[2 n]]; Table[a[n], {n, 0, 14}]

a(n) = numerator(bernfrac(2*n)*(2*n)!); \\ Michel Marcus, Sep 01 2021

a(n) is the numerator of Bernoulli(2*n) * (2*n)! (for denominators see A128059).
a(n) is the numerator of (2*n)!^2 * [x^(2*n)] x * coth(x/2) / 2.
a(n) is the numerator of b(2*n) where b(n) = -Sum_{k=1..n} binomial(n,k)^2 * k! * b(n-k) / (k+1), b(0) = 1.

A273878 Numerator of (2*(n+1)!/(n+2)).

Original entry on oeis.org

1, 4, 3, 48, 40, 1440, 1260, 8960, 72576, 7257600, 6652800, 958003200, 889574400, 11623772160, 163459296000, 41845579776000, 39520825344000, 12804747411456000, 12164510040883200, 231704953159680000, 4644631106519040000

Offset: 0

Johannes W. Meijer, Jun 08 2016

The moments, i.e. E(X^n) = int(x^n * p(x), x = 0..infinity) for n > 0, of the probability density function p(x) = 2*x*E(x, 1, 1), see A163931, lead to this sequence.

			The first few moments of p(x) are: 1, 4/3, 3, 48/5, 40, 1440/7, … .

J. W. Meijer and N. H. G. Baken, The Exponential Integral Distribution, Statistics and Probability Letters, Volume 5, No.3, April 1987. pp 209-211.

Cf. A090585 (denominators), A090586, A085250, A110468, A128059, A128060, A163931.

a := proc(n): numer(2*(n+1)!/(n+2)) end: seq(a(n), n=0..20);

a(n) = numerator(2*(n+1)!/(n+2)) \\ Felix Fröhlich, Jun 09 2016

a(n) = numer(2*(n+1)!/(n+2))
a(n) = (n+1) * A090586(n+1)
a(2*n) = A110468(n) and a(2*n+1) = (2*n)!*A085250(n+1)/A128060(n+2).

cp's OEIS Frontend

A356360 Denominator of the continued fraction 1/(2-3/(3-4/(4-5/(...(n-1)-n/(n+1))))).

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A128060 a(n) = 2*n - numerator((2*n-1)^2/(2*(2*n)!)).

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A145737 a(n) = square part of A145609(n).

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A145738 a(n) = squarefree part of A145609(n).

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A347425 a(n) = Bernoulli(2*n) * (2*n+1)! if 2*n+1 is a prime, otherwise a(n) = Bernoulli(2*n) * (2*n)!.

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A273878 Numerator of (2*(n+1)!/(n+2)).

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PARI

Formula

A128060 a(n) = 2n - numerator((2n-1)^2/(2(2n)!)).

A347425 a(n) = Bernoulli(2n) (2n+1)! if 2n+1 is a prime, otherwise a(n) = Bernoulli(2n) (2*n)!.