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A038507
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n! + 1.
(Formerly N0107)
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37
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2, 2, 3, 7, 25, 121, 721, 5041, 40321, 362881, 3628801, 39916801, 479001601, 6227020801, 87178291201, 1307674368001, 20922789888001, 355687428096001, 6402373705728001, 121645100408832001
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,1
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COMMENTS
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"For n = 4, 5 and 7, n!+1 is a square. Sierpinski 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.
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REFERENCES
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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
C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, 1966, p. 82.
Waclaw Sierpinski, 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).
Arthur T. White, Ringing the changes, Math. Proc. Cambridge Philos. Soc. 94 (1983), no. 2, 203-215.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..300
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 763
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 834
T. Mansour and J. West, Avoiding 2-letter signed patterns.
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
R. G. Wilson v, Explicit factorizations
Index entries for sequences related to factorial numbers
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FORMULA
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From Sergei N. Gladkovskii, Jun 22 2012: (Start)
G.f.: A(x) = Q(0) where Q(k) = 1 + 1/(k! - k!*(k+1)!*x/((k+1)!*x + 1/Q(k+1))); (continued fraction, 3rd kind, 3-step).
Let E(x)=1/(1-x) + exp(x) be the e.g.f., then
E(x) = Q(0) where Q(k) = 1 + 1/(k! - k!*(k+1)!*x/((k+1)!*x + (k+1)/Q(k+1))); (continued fraction, 3rd kind, 3-step).
(End).
a(n) = n * (a(n-1) - 1) + 1. - Reinhard Zumkeller, Mar 20 2013
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MATHEMATICA
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Range[0, 20]!+1 (* From Harvey P. Dale, May 06 2012 *)
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PROG
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(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
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CROSSREFS
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Cf. A000142, A033312, A002583, A051301, A056111, A002981.
Sequence in context: A076996 A139148 A185387 * A077001 A180996 A087522
Adjacent sequences: A038504 A038505 A038506 * A038508 A038509 A038510
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane. Entry revised by N. J. A. Sloane, Jun 10 2012
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EXTENSIONS
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Additional comments from Jason Earls (zevi_35711(AT)yahoo.com), Apr 01 2001
Numericana.com URL fixed by Gerard P. Michon, Mar 30 2010
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STATUS
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approved
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