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.
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
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.
Toufik 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: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 (2012).
Hisanori Mishima, Factorizations of many number sequences.
Lara Pudwell, Pattern Avoidance in Circular Parking Functions, Valparaiso Univ. (2026). See pp. 11 (Table 3), 12 (Theorem 14).
Wouter van Doorn and Terence Tao, Growth rates of sequences governed by the squarefree properties of its translates, arXiv:2512.01087 [math.NT], 2025. See p. 2.
Andrew Walker, Factors of n! +- 1 (2001).
Arthur T. White, Ringing the changes, Math. Proc. Cambridge Philos. Soc. (1983) Vol 94, 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.
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
(Python)
from math import factorial
def A038507(n): return factorial(n) + 1 # Karl-Heinz Hofmann, Aug 21 2024
(SageMath) [factorial(n) + 1 for n in range(0, 24)] # Stefano Spezia, Apr 21 2025
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
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