A006862 - OEIS

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A006862

Euclid numbers: 1 + product of the first n primes.
(Formerly M2698)

2, 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, 6469693231, 200560490131, 7420738134811, 304250263527211, 13082761331670031, 614889782588491411, 32589158477190044731, 1922760350154212639071 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,1`
	COMMENTS	`It is an open question whether all terms of this sequence are squarefree.` `a(n) is the smallest x > 1 such that x^prime(n) == 1 (mod prime(i)) i=1,2,3,...,n-1. - Benoit Cloitre, May 30 2002` `Numbers n such that n/phi(n-1) is a record. - Arkadiusz Wesolowski, Nov 22 2012`
	REFERENCES	`J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 211, p. 61, Ellipses, Paris 2008.` `S. W. Golomb, The evidence for Fortune's conjecture, Math. Mag. 54 (1981), 209-210.` `R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990.` `F. Iacobescu, Smarandache partition type and other sequences, Bulletin of pure and applied sciences, Vol. 16E, No. 2, pp. 237-240.` `H. Ibstedt, A Few Smarandache Sequences, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, 170-183.` `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, 2012 - From N. J. A. Sloane, Jun 13 2012` `N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).` `F. Smarandache, Properties of numbers, Arizona State University Special Collections, 1973.` `I. Vardi, Computational Recreations in Mathematica, Addison-Wesley, 1991, sections 5.1 and 5.2.` `S. Wagon, Mathematica in Action, Freeman, NY, 1991, p. 35.`
	LINKS	`T. D. Noe, Table of n, a(n) for n = 0..100` `Hisanori Mishima, Factorizations of many number sequences` `Eric Weisstein's World of Mathematics, Euclid Number` `Eric Weisstein's World of Mathematics, Fortunate Prime` `R. G. Wilson v, Explicit factorizations`
	FORMULA	`a(n) = A002110(n) + 1.`
	MAPLE	`with(numtheory): A006862 := proc(n) local i; if n=0 then 2 else 1+product('ithprime(i)', 'i'=1..n); fi; end;`
	MATHEMATICA	`Table[Product[Prime[k], {k, 1, n}] + 1, {n, 1, 18}]`
	PROG	`(PARI) a(n)=my(v=primes(n)); prod(i=1, #v, v[i])+1 \\ Charles R Greathouse IV, Nov 20 2012`
	CROSSREFS	`Cf. A014545, A057588, A018239 (primes), A005867.` `Sequence in context: A089359 A081947 A046972 * A038710 A073918 A096350` `Adjacent sequences: A006859 A006860 A006861 * A006863 A006864 A006865`
	KEYWORD	`nonn,nice,easy,changed`
	AUTHOR	`Simon Plouffe and N. J. A. Sloane.`
	STATUS	`approved`

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Last modified May 23 02:53 EDT 2013. Contains 225585 sequences.