cp's OEIS Frontend

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

%I A002585 M2697 N1081 #69 Feb 16 2025 08:32:26
%S A002585 3,7,31,211,2311,509,277,27953,703763,34231,200560490131,676421,
%T A002585 11072701,78339888213593,13808181181,18564761860301,19026377261,
%U A002585 525956867082542470777,143581524529603,2892214489673,16156160491570418147806951,96888414202798247,1004988035964897329167431269
%N A002585 Largest prime factor of 1 + (product of first n primes).
%C A002585 Based on Euclid's proof that there are infinitely many primes.
%C A002585 The products of the first primes are called primorial numbers. - _Franklin T. Adams-Watters_, Jun 12 2014
%D A002585 M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors).
%D A002585 M. Kraitchik, Introduction à la Théorie des Nombres. Gauthier-Villars, Paris, 1952, p. 2.
%D A002585 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002585 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002585 <a href="/A002585/b002585.txt">Table of n, a(n) for n = 1..98</a>
%H A002585 A. Borning, <a href="http://dx.doi.org/10.1090/S0025-5718-1972-0308018-5 ">Some results for k!+-1 and 2.3.5...p+-1</a>, Math. Comp., 26 (1972), 567-570.
%H A002585 M. Kraitchik, <a href="/A002582/a002582.pdf">On the divisibility of factorials</a>, Scripta Math., 14 (1948), 24-26 (but beware errors). [Annotated scanned copy]
%H A002585 S. Kravitz and D. E. Penney, <a href="http://www.jstor.org/stable/2689826">An extension of Trigg's table</a>, Math. Mag., 48 (1975), 92-96.
%H A002585 S. Kravitz and D. E. Penney, <a href="/A002584/a002584.pdf">An extension of Trigg's table</a>, Math. Mag., 48 (1975), 92-96. [Annotated scanned copy; also letter from N. J. A. Sloane to John Selfridge]
%H A002585 R. Mestrovic, <a href="http://arxiv.org/abs/1202.3670">Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof</a>, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2018. - From _N. J. A. Sloane_, Jun 13 2012
%H A002585 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha102.htm">Factorizations of many number sequences</a>
%H A002585 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha103.htm">Factorizations of many number sequences</a>
%H A002585 John Selfridge, Marvin Wunderlich, Robert Morris, N. J. A. Sloane, <a href="/A002584/a002584_1.pdf">Correspondence, 1975</a>
%H A002585 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EuclidNumber.html">Euclid Number.</a>
%H A002585 R. G. Wilson v, <a href="/A038507/a038507.txt">Explicit factorizations</a>
%F A002585 a(n) = A006530(A006862(n)). - _Amiram Eldar_, Feb 13 2020
%t A002585 FactorInteger[#][[-1,1]]&/@Rest[FoldList[Times,1,Prime[Range[30]]]+1] (* _Harvey P. Dale_, Apr 10 2012 *)
%o A002585 (PARI) a(n)=my(f=factor(prod(i=1,n,prime(i))+1)[,1]); f[#f] \\ _Charles R Greathouse IV_, Feb 07 2017
%Y A002585 Cf. A002110, A002584, A006530, A006862, A051342.
%K A002585 nonn,nice
%O A002585 1,1
%A A002585 _N. J. A. Sloane_
%E A002585 More terms from _Labos Elemer_, May 02 2000
%E A002585 More terms from _Robert G. Wilson v_, Mar 24 2001
%E A002585 Terms up to a(81) in b-file added by _Sean A. Irvine_, Apr 19 2014
%E A002585 Terms a(82)-a(87) in b-file added by _Amiram Eldar_, Feb 13 2020
%E A002585 Terms a(88)-a(98) in b-file added by _Max Alekseyev_, Aug 26 2021