A129924 -id:A129924 - cp's OEIS frontend

A061354 Numerator of Sum_{k=0..n} 1/k!.

1, 2, 5, 8, 65, 163, 1957, 685, 109601, 98641, 9864101, 13563139, 260412269, 8463398743, 47395032961, 888656868019, 56874039553217, 7437374403113, 17403456103284421, 82666416490601, 6613313319248080001, 69439789852104840011

Offset: 0

Amarnath Murthy, Apr 28 2001

p divides a(p-1) for prime p = {2, 5, 13, 37, 463, ...} which apparently coincides with A064384(n) = {2, 5, 13, 37, 463, ...} Primes p such that p divides 0!-1!+2!-3!+...+(-1)^{p-1}(p-1)!. - Alexander Adamchuk, Jun 14 2007
GCD(a(n), a(n+2)) = A124779(n) is either 1 or a prime 2, 5, 13, 37, 463, ... = A064384. - Jonathan Sondow, Jun 12 2007
For proofs of Adamchuk's and my Comments, see the link "The Taylor series for e ...". - Jonathan Sondow, Jun 18 2007

			1, 2, 5/2, 8/3, 65/24, 163/60, 1957/720, 685/252, ...

Harry J. Smith, Table of n, a(n) for n = 0..200
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006), 637-641.
J. Sondow, The Taylor series for e and the primes 2, 5, 13, 37, 463, ...: a surprising connection
J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.
Index entries for sequences related to factorial numbers

Cf. A061355 (denominators), A093101, A064384, A064384, A124779, A129924.

exp[n_]:=Apply[Plus,1/Range[0,n]!];Numerator[Table[exp[n],{n,0,21}]]  (* Geoffrey Critzer, May 05 2013 *)
A061354[n_] := Numerator[Sum[1/k!, {k, 0, n}]]; Array[A061354, 22, 0] (* JungHwan Min, Nov 08 2016 *)
Accumulate[1/Range[0,30]!]//Numerator (* Harvey P. Dale, Apr 13 2018 *)

{ default(realprecision, 500); e=exp(1); for (n=0, 200, a=numerator(floor(n!*e)/n!); if (n==0, a=1); write("b061354.txt", n, " ", a) ) } \\ Harry J. Smith, Jul 21 2009

a(n) = A000522(n)/A093101(n).
Numerators of floor(n!*exp(1))/n!, n>=1. Numerators of coefficients in expansion of exp(x)/(1-x). - Vladeta Jovovic, Aug 11 2002
a(n) = (1+n+n(n-1)+...+n!)/GCD(n!,1+n+n(n-1)+...+n!). - Jonathan Sondow, Aug 18 2006

A064384 Primes p such that p divides 0!-1!+2!-3!+...+(-1)^{p-1}(p-1)!.

Original entry on oeis.org

2, 5, 13, 37, 463

Offset: 1

Kevin Buzzard (buzzard(AT)ic.ac.uk), Sep 28 2001

If p is in the sequence then p divides 0!-1!+2!-3!+...+(-1)^N N! for all sufficiently large N. Naive heuristics suggest that the sequence should be infinite but very sparse.
Same as the terms > 1 in A124779. - Jonathan Sondow, Nov 09 2006
A prime p is in the sequence if and only if p|A(p-1), where A(0) = 1 and A(n) = n*A(n-1)+1 = A000522(n). - Jonathan Sondow, Dec 22 2006
Also, a prime p is in this sequence if and only if p divides A061354(p-1). - Alexander Adamchuk, Jun 14 2007
Michael Mossinghoff has calculated that 2, 5, 13, 37, 463 are the only terms up to 150 million. - Jonathan Sondow, Jun 12 2007

			5 is in the sequence because 5 is prime and it divides 0!-1!+2!-3!+4!=20.

R. K. Guy, Unsolved Problems in Theory of Numbers, Springer-Verlag, Third Edition, 2004, B43.

Jonathan Sondow, The Taylor series for e and the primes 2, 5, 13, 37, 463: a surprising connection
Jonathan Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, arXiv:0709.0671 [math.NT], 2007-2009.
Index entries for sequences related to factorial numbers

Cf. A064383, A124779, A000522, A061354, A129924.

Select[Select[Range[500], PrimeQ], (Mod[Sum[(-1)^(p - 1)*p!, {p, 2, # - 1}], #] == 0) &] (* Julien Kluge, Feb 13 2016 *)
a[0] = 1; a[n_] := a[n] = n*a[n - 1] + 1; Select[Select[Range[500], PrimeQ], (Mod[a[# - 1], #] == 0) &] (* Julien Kluge, Feb 13 2016 with the sequence approach suggested by Jonathan Sondow *)
Select[Prime[Range[500]],Divisible[AlternatingFactorial[#]-1,#]&] (* Harvey P. Dale, Jan 08 2021 *)

A=1;for(n=1,1000,if(isprime(n),if(Mod(A,n)==0,print(n)));A=n*A+1) \\ Jonathan Sondow, Dec 22 2006

Edited by Max Alekseyev, Mar 05 2011

A124779 a(n) = gcd(A(n), A(n+2))/gcd(d(n), d(n+2)) where A(n) = Sum_{k=0..n} n!/k! and d(n) = gcd(A(n), n!).

Original entry on oeis.org

1, 2, 5, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 37, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Jonathan Sondow, Nov 07 2006

nonn

The next term > 1 is a(460) = 463. The primes 2, 5, 13, 37, 463 are the only terms > 1 up to n = 600000. If a(n) > 1 with n > 1, then a(n) = n+3 is prime. This uses A(n+2) = (n+2)(n+1)*A(n) + n+3. The terms > 1 are A064384 = primes p such that p divides 0!-1!+2!-3!+...+(-1)^{p-1}(p-1)!. The proof uses (n-1)!/(n-k-1)! = (n-1)(n-2)...(n-k) == (-1)^k k! (mod n). Cf. Cloitre's comment in A064383.
An integer p > 1 is in the sequence if and only if p is prime and p|A(p-1), where A(0) = 1 and A(n) = n*A(n-1)+1 for n > 0. - Jonathan Sondow, Dec 22 2006
Michael Mossinghoff has calculated that there are only five primes in the sequence up to 150 million. Heuristics suggest it contains infinitely many. - Jonathan Sondow, Jun 12 2007

			a(2) = gcd(A(2), A(4))/gcd(d(2), d(4)) = gcd(5, 65)/gcd(1, 1) = 5/1 = 5.

R. K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, 3rd edition, 2004, B43.

J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, arXiv:0704.1282 [math.HO], 2007-2010.
J. Sondow, The Taylor series for e and the primes 2, 5, 13, 37, 463: a surprising connection
J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.
Eric Weisstein's World of Mathematics, Alternating Factorial
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Index entries for sequences related to factorial numbers

A(n) = A000522, d(n) = A093101, gcd(A(n), A(n+2)) = A124780, gcd(d(n), d(n+2)) = A124781, (n+3)/gcd(A(n), A(n+2)) = A124782, (n+3)/gcd(d(n), d(n+2)) = A123901. Cf. A061354, A061355, A123899, A123900.

Cf. A129924.

(A[n_] := Sum[n!/k!, {k, 0, n}]; d[n_] := GCD[A[n],n! ]; Table[GCD[A[n],A[n+2]]/GCD[d[n],d[n+2]], {n,0,100}])

A124779(n)={my(An=A000522(n),A2=A000522(n+2));gcd(An, A2)/gcd([An,n!,A2,(n+2)!])} \\ M. F. Hasler, Jun 04 2019

a(n) = A124780(n)/A124781(n) = A124782(n)/A123901(n).
a(n) = gcd(A(n), A(n+2))/gcd(A(n), A(n+2), n!) where A(n)=1+n+n(n-1)+...+n!. - Jonathan Sondow, Nov 10 2006
a(n) = gcd(N(n), N(n+2)), where N(n) = A061354(n) = numerator of Sum[1/k!,{k,0,n}]. - Jonathan Sondow, Jun 12 2007

A064383 Integers n >= 1 such that n divides 0!-1!+2!-3!+4!-...+(-1)^{n-1}(n-1)!.

Original entry on oeis.org

1, 2, 4, 5, 10, 13, 20, 26, 37, 52, 65, 74, 130, 148, 185, 260, 370, 463, 481, 740, 926, 962, 1852, 1924, 2315, 2405, 4630, 4810, 6019, 9260, 9620, 12038, 17131, 24076, 30095, 34262, 60190, 68524, 85655, 120380, 171310, 222703, 342620, 445406, 890812, 1113515

Offset: 1

Kevin Buzzard (buzzard(AT)ic.ac.uk), Sep 28 2001

If a is in the sequence, then so are all its positive divisors. If a and b are coprime and in the sequence, then so is their product. Hence in extending the sequence, one may as well just look for primes in the sequence (and then check powers of these primes). Heuristically one might expect a very sparse but infinite set of primes in the sequence, but the largest one I know is p=463 and I've searched up to 600000. This sequence was brought to my attention by David Loeffler.
Also, n such that A000522(n)==1 (mod n^2). - Benoit Cloitre, Apr 15 2003
The primes in this sequence are the same as the terms > 1 in A124779. - Jonathan Sondow, Nov 09 2006
Also, n such that n|A(n-1), where A(0) = 1 and A(k) = k*A(k-1)+1 = A000522(k) for k > 0. - Jonathan Sondow, Dec 22 2006
Michael Mossinghoff has calculated that 2, 5, 13, 37, 463 are the only primes in the sequence up to 150 million. - Jonathan Sondow, Jun 12 2007

			4 is in the sequence because 4 divides 0!-1!+2!-3!=1-1+2-6=-4.

R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer-Verlag, 2004, B43.

J. Sondow, The Taylor series for e and the primes 2, 5, 13, 37, 463: a surprising connection
J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.
Index entries for sequences related to factorial numbers

Cf. A000522, A057245, A064384, A124779, A129924.

s = 0; Do[ s = s + (-1)^(n)(n)!; If[ Mod[ s, n + 1 ] == 0, Print[ n + 1 ] ], {n, 0, 600000} ]
Divisors[4454060] (* From Formula above *) (* Harvey P. Dale, Aug 09 2012 *)

Up to n=600000, these are just the divisors of 4*5*13*37*463.

More terms from Sean A. Irvine, Jul 02 2023

cp's OEIS Frontend

A061354 Numerator of Sum_{k=0..n} 1/k!.

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A064384 Primes p such that p divides 0!-1!+2!-3!+...+(-1)^{p-1}(p-1)!.

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A124779 a(n) = gcd(A(n), A(n+2))/gcd(d(n), d(n+2)) where A(n) = Sum_{k=0..n} n!/k! and d(n) = gcd(A(n), n!).

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A064383 Integers n >= 1 such that n divides 0!-1!+2!-3!+4!-...+(-1)^{n-1}(n-1)!.

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