

A001272


Numbers n such that n!  (n1)! + (n2)!  (n3)! + ...  (1)^n*1! is prime.


8



3, 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160, 661, 2653, 3069, 3943, 4053, 4998, 8275, 9158, 11164, 43592, 59961
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OFFSET

1,1


COMMENTS

At present the terms greater than or equal to 2653 are only probable primes.
Živković shows that all terms must be less than p = 3612703, which divides a(n) for n >= p.  T. D. Noe, Jan 25 2008
Notwithstanding Živković's wording, p = 3612703 also divides the alternating factorial for n = 3612702. [Guy: If there is a value of n such that n + 1 divides af(n), then n + 1 will divide af(m) for all m > n.] Therefore af(3612701), approximately 7.3 * 10^22122513, is the final primality candidate.  Hans Havermann, Jun 17 2013
Next term (if it exists) has n > 100000 (per M. Rodenkirch post).  Eric W. Weisstein, Dec 18 2017


REFERENCES

J.M. De Koninck, Ces nombres qui nous fascinent, Entry 160, p. 52, Ellipses, Paris 2008.
Martin Gardner, Strong Laws of Small Primes, in The Last Recreations, p. 198 (1997).
R. K. Guy, Unsolved Problems in Number Theory, B43.


LINKS

Table of n, a(n) for n=1..25.
factordb.com, Primality at factordatabase of the alternating factorial for n=661
Rudolf Ondrejka, The Top Ten: a Catalogue of Primal Configurations
Rodenkirch, M. Alternating Factorials
Eric Weisstein's World of Mathematics, Alternating Factorial
Eric Weisstein's World of Mathematics, Factorial Sums
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Miodrag Živković, The number of primes Sum_{i=1..n} (1)^(ni)*i! is finite, Math. Comp. 68 (1999), no. 225, 403409.
Index entries for sequences related to factorial numbers


MAPLE

with(numtheory); f := proc(n) local i; add((1)^(ni)*i!, i=1..n); end; isprime(f(15));


MATHEMATICA

(* This program is not convenient for more than 15 terms *) Reap[For[n = 1, n <= 1000, n++, If[PrimeQ[Sum[(1)^(n  k) * k!, {k, n}]], Print[n]; Sow[n]]]][[2, 1]] (* JeanFrançois Alcover, Sep 05 2013 *)
Position[AlternatingFactorial[Range[200]], _?PrimeQ] // Flatten (* Eric W. Weisstein, Sep 19 2017 *)


CROSSREFS

Cf. A005165, A002981, A002982, A100289.
Sequence in context: A066378 A125684 A207669 * A273664 A047563 A261604
Adjacent sequences: A001269 A001270 A001271 * A001273 A001274 A001275


KEYWORD

nonn,hard,more,nice,fini


AUTHOR

N. J. A. Sloane


EXTENSIONS

661 found independently by Eric W. Weisstein and Rachel Lewis (racheljlewis(AT)hotmail.com); 2653 and 3069 found independently by Chris Nash (nashc(AT)lexmark.com) and Rachel Lewis (racheljlewis(AT)hotmail.com)
3943, 4053, 4998 found by Paul Jobling (paul.jobling(AT)whitecross.com)
8275, 9158 found by team of Rachel Lewis, Paul Jobling and Chris Nash
661!  660! + 659!  ... was shown to be prime by team of Giovanni La Barbera and others using the Certifix program developed by Marcel Martin, Jul 15 2000 (see link)  Paul Jobling (Paul.Jobling(AT)WhiteCross.com) and Giovanni La Barbera, Aug 02 2000
a(23) = 11164 found by Paul Jobling, Nov 25 2004
Edited by T. D. Noe, Oct 30 2008
Edited by Hans Havermann, Jun 17 2013
a(24) = 43592 from Serge Batalov, Jul 19 2017
a(25) = 59961 from Mark Rodenkirch, Sep 18 2017


STATUS

approved



