A001272 Numbers k such that k! - (k-1)! + (k-2)! - (k-3)! + ... - (-1)^k*1! is prime.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• more
• nice
• fini

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.
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 97.

Links

• factordb.com, Primality at factor-database of the alternating factorial for n=661.
• Rudolf Ondrejka, The Top Ten: a Catalogue of Primal Configurations.
• M. Rodenkirch, 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)^(n-i)*i! is finite, Math. Comp. 68 (1999), no. 225, 403-409.
• Index entries for sequences related to factorial numbers.

Crossrefs

Cf. A005165, A002981, A002982, A100289.

Programs

Maple

with(numtheory); f := proc(n) local i; add((-1)^(n-i)*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]] (* Jean-François Alcover, Sep 05 2013 *)
Position[AlternatingFactorial[Range[200]], ?PrimeQ] // Flatten (* _Eric W. Weisstein, Sep 19 2017 *)

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