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A006862 Euclid numbers: 1 + product of the first n primes.
(Formerly M2698)
46
2, 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, 6469693231, 200560490131, 7420738134811, 304250263527211, 13082761331670031, 614889782588491411, 32589158477190044731, 1922760350154212639071, 117288381359406970983271, 7858321551080267055879091 (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

Nyblom (theorem 2.3) proves that this sequence contains no proper powers, e.g., is a subsequence of A007916. - Charles R Greathouse IV, Mar 02 2016

It is an open question if there are an infinite number of prime Euclid numbers. - Mike Winkler, Feb 05 2017

These numbers are not pairwise relatively prime; the first example is gcd(a(7), a(17)) = 277. Also gcd(a(47), a(131)) = 1051, which is probably the second example (wrt. greater index which is here 131). It is easy to find other primes like 277 and 1051. - Jeppe Stig Nielsen, Mar 24 2017

REFERENCES

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 211, p. 61, Ellipses, Paris 2008.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990.

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

S. W. Golomb, The evidence for Fortune's conjecture, Math. Mag. 54 (1981), 209-210.

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 [math.HO], 2012.

Hisanori Mishima, Factorizations of many number sequences

Michael A. Nyblom, On the construction of a family of almost power free sequences, Fibonacci Quart. 46/47 (2008/2009), no. 4, 366-368.

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

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.

EXAMPLE

It is a universal convention that an empty product is 1 (just as an empty sum is 0), and since this sequence has offset 0, the first term is 1+1 = 2. - N. J. A. Sloane, Dec 02 2015

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}]

1 + FoldList[Times, 1, Prime@ Range@ 19] (* Harvey P. Dale, Dec 02 2015 and modified by Robert G. Wilson v, Mar 25 2017 *)

PROG

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

(MAGMA) [2] cat [&*PrimesUpTo(p)+1: p in PrimesUpTo(70)]; // Vincenzo Librandi, Dec 03 2015

CROSSREFS

Cf. A014545, A057588, A018239 (primes), A005867, A007916.

Sequence in context: A267487 A081947 A046972 * A038710 A241196 A073918

Adjacent sequences:  A006859 A006860 A006861 * A006863 A006864 A006865

KEYWORD

nonn,nice,easy

AUTHOR

Simon Plouffe and N. J. A. Sloane

STATUS

approved

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Last modified June 26 15:11 EDT 2017. Contains 288766 sequences.