

A001272


Numbers k such that k!  (k1)! + (k2)!  (k3)! + ...  (1)^k*1! is prime.


10



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 the alternating factorial af(k) for k >= p.  T. D. Noe, Jan 25 2008
Notwithstanding Živković's wording, p = 3612703 also divides the alternating factorial for k = 3612702. [Guy: If there is a value of k such that k + 1 divides af(k), then k + 1 will divide af(m) for all m > k.] Therefore af(3612701), approximately 7.3 * 10^22122513, is the final primality candidate.  Hans Havermann, Jun 17 2013
Next term (if it exists) has k > 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



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



KEYWORD

nonn,hard,more,nice,fini


AUTHOR



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


STATUS

approved



