A090585 -id:A090585 - cp's OEIS frontend

A174530 Numerators of the second row of the Akiyama-Tanigawa table for the sequence 1/n!.

-1, 0, 3, 4, 5, 1, 7, 1, 1, 1, 11, 1, 13, 1, 1, 1, 17, 1, 19, 1, 1, 1, 23, 1, 1, 1, 1, 1, 29, 1, 31, 1, 1, 1, 1, 1, 37, 1, 1, 1, 41, 1, 43, 1, 1, 1, 47, 1, 1, 1, 1, 1, 53, 1, 1, 1, 1, 1, 59, 1, 61, 1, 1, 1, 1, 1, 67, 1, 1, 1, 71, 1, 73, 1, 1, 1, 1, 1, 79

Offset: 0

Paul Curtz, Mar 21 2010

Filling the top row of a table with T(0,k) = 1/k!, k>=0, the Akiyama-Tanigawa algorithm constructs the following table T(n,k) of fractions, n>=0, k>=0:
1, 1, 1/2, 1/6, 1/24, 1/120, 1/720, 1/5040, 1/40320, 1/362880,...
0, 1, 1, 1/2, 1/6, 1/24, 1/120, 1/720, 1/5040, 1/40320, 1/362880, ...
-1, 0, 3/2, 4/3, 5/8, 1/5, 7/144, 1/105, 1/640, 1/4536, 11/403200, ...
-1, -3, 1/2, 17/6, 17/8, 109/120, 197/720, 107/1680, 487/40320, ..
2, -7, -7, 17/6, 73/12, 457/120, 529/360, 2081/5040, 263/2880,...
9, 0, -59/2, -13, 91/8, 421/30, 355/48, 2161/840, 3871/5760, 709/5040, ..
9, 59, -99/2, -195/2, -319/24, 1593/40, 2701/80, 76631/5040, 4285/896,...
The numerators of T(2,k) are the current sequence.
The denominators are 1, 1, 2, 3, 8, 5, 144, 105, 640, 4536, 403200, 332640, 43545600, 37065600,...
T(0,k) = T(1,k+1), shifted.
The left column is T(n,0) = (-1)^(n+1)*A014182(n).
The column T(n,1) appears to be (-1)^n*A074051(n). - R. J. Mathar, Jan 16 2011
a(n) = numerator(A005563(n-1)/(n-1)!), for n>0. - Fred Daniel Kline, Mar 20 2016

D. Merlini, R. Sprugnoli, M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05.

Cf. A089026, A090585, A080305.

nn = 78; Numerator[Simplify[CoefficientList[Series[-Zeta[x] + (Derivative[1][Zeta][x] + x*Derivative[2][Zeta][x])*x, {x, 0, nn}], x]/Table[Derivative[n][Zeta][0], {n, 0, nn}]]] (* Mats Granvik, Nov 11 2013 *)

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).

A308090 a(n) = gcd(2^n + n!, 3^n + n!, n+1).

Original entry on oeis.org

1, 1, 1, 5, 1, 7, 1, 1, 1, 11, 1, 13, 1, 1, 1, 17, 1, 19, 1, 1, 1, 23, 1, 1, 1, 1, 1, 29, 1, 31, 1, 1, 1, 1, 1, 37, 1, 1, 1, 41, 1, 43, 1, 1, 1, 47, 1, 1, 1, 1, 1, 53, 1, 1, 1, 1, 1, 59, 1, 61, 1, 1, 1, 1, 1, 67, 1, 1, 1, 71, 1, 73, 1, 1, 1, 1, 1, 79, 1, 1, 1, 83, 1, 1, 1, 1, 1, 89, 1, 1, 1, 1, 1, 1, 1, 97, 1, 1, 1

Offset: 1

Pedro Caceres, May 11 2019

nonn

From observation: For n > 3, if n+1 is prime, then a(n) = n+1.
This implies that (2^n + n!)= 0 mod (n+1) iff (n+1) is prime, and (3^n + n!)= 0 mod (n+1) iff (n+1) is prime.
Conjecture: Conversely, if gcd(2^n + n!, 3^n + n!, n+1) = n+1, then n+1 is prime.
Appears to be the same as A090585(n) except at n=2. - R. J. Mathar, Jul 22 2021

			a(4) = gcd(2^4 + 4!, 3^4 + 4!, 5) = gcd(40, 105, 5) = 5.
a(5) = gcd(2^5 + 5!, 3^5 + 5!, 6) = gcd(152, 363, 6) = 1.

Cf. A007611, A249945.

Table[GCD[2^n+n!,3^n+n!,n+1],{n,100}] (* Harvey P. Dale, Aug 27 2020 *)

a(n) = gcd([2^n + n!, 3^n + n!, n+1]); \\ Michel Marcus, May 12 2019

a(n) = gcd(A007611(n), A249945(n), n+1).

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A174530 Numerators of the second row of the Akiyama-Tanigawa table for the sequence 1/n!.

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

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A308090 a(n) = gcd(2^n + n!, 3^n + n!, n+1).

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