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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)



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


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.

Shubhankar Paul, Ten Problems of Number Theory, International Journal of Engineering and Technical Research (IJETR), ISSN: 2321-0869, Volume-1, Issue-9, November 2013

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.


T. D. Noe, Table of n, a(n) for n = 0..100

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

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


a(n) = A002110(n) + 1.


with(numtheory): A006862 := proc(n) local i; if n=0 then 2 else 1+product('ithprime(i)', 'i'=1..n); fi; end;


Table[Product[Prime[k], {k, 1, n}] + 1, {n, 1, 18}]


(PARI) a(n)=my(v=primes(n)); prod(i=1, #v, v[i])+1 \\ Charles R Greathouse IV, Nov 20 2012


Cf. A014545, A057588, A018239 (primes), A005867.

Sequence in context: A239892 A081947 A046972 * A038710 A241196 A073918

Adjacent sequences:  A006859 A006860 A006861 * A006863 A006864 A006865




Simon Plouffe and N. J. A. Sloane.



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Last modified October 30 13:32 EDT 2014. Contains 248803 sequences.