A002585 Largest prime factor of 1 + (product of first n primes). (Formerly M2697 N1081)

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

Offset

1,1

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

Formula

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A123332 A343087 A051342 * A365021 A103785 A289127

Adjacent sequences: A002582 A002583 A002584 * A002586 A002587 A002588

Keyword

nonn,nice

Author

N. J. A. Sloane

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

Status

approved