A038507 a(n) = n! + 1. (Formerly N0107)

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

Offset

0,1

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.

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.

R. 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, 2012 - From N. J. A. Sloane, Jun 13 2012

G. 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

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)

Example

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A139148 A293246 A185387 * A077001 A180996 A307503

Adjacent sequences: A038504 A038505 A038506 * A038508 A038509 A038510

Keyword

nonn,easy,nice,changed

Author

N. J. A. Sloane

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

Status

approved