A002585 Largest prime factor of 1 + (product of first n primes).

3, 7, 31, 211, 2311, 509, 277, 27953, 703763, 34231, 200560490131, 676421, 11072701, 78339888213593, 13808181181, 18564761860301, 19026377261, 525956867082542470777, 143581524529603, 2892214489673, 16156160491570418147806951, 96888414202798247, 1004988035964897329167431269

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

Based on Euclid's proof that there are infinitely many primes.

The products of the first primes are called primorial numbers. - Franklin T. Adams-Watters, Jun 12 2014

References

• M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors).
• M. Kraitchik, Introduction à la Théorie des Nombres. Gauthier-Villars, Paris, 1952, p. 2.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Table of n, a(n) for n = 1..98
• A. Borning, Some results for k!+-1 and 2.3.5...p+-1, Math. Comp., 26 (1972), 567-570.
• M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors). [Annotated scanned copy]
• S. Kravitz and D. E. Penney, An extension of Trigg's table, Math. Mag., 48 (1975), 92-96.
• S. Kravitz and D. E. Penney, An extension of Trigg's table, Math. Mag., 48 (1975), 92-96. [Annotated scanned copy; also letter from N. J. A. Sloane to John Selfridge]
• 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 [math.HO], 2012-2018. - From _N. J. A. Sloane_, Jun 13 2012
• Hisanori Mishima, Factorizations of many number sequences
• Hisanori Mishima, Factorizations of many number sequences
• John Selfridge, Marvin Wunderlich, Robert Morris, N. J. A. Sloane, Correspondence, 1975
• Eric Weisstein's World of Mathematics, Euclid Number.
• R. G. Wilson v, Explicit factorizations

Crossrefs

Cf. A002110, A002584, A006530, A006862, A051342.

Programs

Mathematica

FactorInteger[#][[-1,1]]&/@Rest[FoldList[Times,1,Prime[Range[30]]]+1] (* Harvey P. Dale, Apr 10 2012 *)

PARI

a(n)=my(f=factor(prod(i=1,n,prime(i))+1)[,1]); f[#f] \\ Charles R Greathouse IV, Feb 07 2017

Formula

a(n) = A006530(A006862(n)). - Amiram Eldar, Feb 13 2020

Extensions

More terms from Labos Elemer, May 02 2000

More terms from Robert G. Wilson v, Mar 24 2001

Terms up to a(81) in b-file added by Sean A. Irvine, Apr 19 2014

Terms a(82)-a(87) in b-file added by Amiram Eldar, Feb 13 2020

Terms a(88)-a(98) in b-file added by Max Alekseyev, Aug 26 2021