A038507 a(n) = n! + 1.

2, 2, 3, 7, 25, 121, 721, 5041, 40321, 362881, 3628801, 39916801, 479001601, 6227020801, 87178291201, 1307674368001, 20922789888001, 355687428096001, 6402373705728001, 121645100408832001, 2432902008176640001, 51090942171709440001, 1124000727777607680001, 25852016738884976640001

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

"For n = 4, 5 and 7, n!+1 is a square. Sierpiński asked if there are any other values of n with this property." p. 82 of Ogilvy and Anderson (see A146968).

Number of {12,12*,1*2,21*,2*1}-avoiding signed permutations in the hyperoctahedral group.

After Wilson's Theorem: if (n+1) is prime then (n+1) is the smallest prime factor of a(n). - Karl-Heinz Hofmann, Aug 21 2024

Examples

			G.f. = 2 + 2*x + 3*x^2 + 7*x^3 + 25*x^4 + 121*x^5 + 721*x^6 + 5041*x^7 + ...
		

References

• C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, 1966, p. 82.
• Wacław Sierpiński, On some unsolved problems of arithmetics, Scripta Mathematica, vol. 25 (1960), p. 125.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 763 and Encyclopedia of Combinatorial Structures 834
• T. Mansour and J. West, Avoiding 2-letter signed patterns, arXiv:math/0207204 [math.CO], 2002.
• Romeo Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2023. - From _N. J. A. Sloane_, Jun 13 2012
• Gerard P. Michon, Wilson's Theorem
• Hisanori Mishima, Factorizations of many number sequences
• Hisanori Mishima, Factorizations of many number sequences
• Andrew Walker, Factors of n! +- 1
• Arthur T. White, Ringing the changes, Math. Proc. Cambridge Philos. Soc. 94 (1983), no. 2, 203-215.
• Robert G. Wilson v, Explicit factorizations
• Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 4.
• Index entries for sequences related to factorial numbers

Crossrefs

Cf. A000142, A002583, A002981, A033312, A051301, A056111, A146968, A217702.

Programs

Haskell

a038507 = (+ 1) . a000142
a038507_list = 2 : f 1 2 where
   f x y = z : f (x + 1) z where z = x * (y - 1) + 1
-- Reinhard Zumkeller, Mar 20 2013

Magma

[Factorial(n) +1: n in [0..25]]; // Vincenzo Librandi, Jul 20 2011

Mathematica

Range[0,20]!+1 (* Harvey P. Dale, May 06 2012 *)

Maxima

A038507(n):= n!+1$
makelist(A038507(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

PARI

a(n)=n!+1 \\ Charles R Greathouse IV, Nov 20 2012

Python

from math import factorial
def A038507(n): return factorial(n) + 1 # Karl-Heinz Hofmann, Aug 21 2024

Sage

[factorial(n) + 1 for n in range(0,24)] # Stefano Spezia, Apr 21 2025

Formula

a(n) = n * (a(n-1) - 1) + 1. - Reinhard Zumkeller, Mar 20 2013

0 = a(n)*(a(n+1) - 5*a(n+2) + 5*a(n+3) - a(n+4)) + a(n+1)*(a(n+1) + a(n+2) - 6*a(n+3) + 2*a(n+4)) + a(n+2)*(3*a(n+2) - a(n+3) - a(n+4)) + a(n+3)*(a(n+3)) if n>=0. - Michael Somos, Apr 23 2014

From Ilya Gutkovskiy, Jan 20 2017: (Start)

E.g.f: exp(x) + 1/(1 - x).

Sum_{n>=0} 1/a(n) = A217702. (End)

Extensions

Additional comments from Jason Earls, Apr 01 2001

Numericana.com URL fixed by Gerard P. Michon, Mar 30 2010

Entry revised by N. J. A. Sloane, Jun 10 2012