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A006862
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Euclid numbers: 1 + product of the first n primes.
(Formerly M2698)
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66
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2, 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, 6469693231, 200560490131, 7420738134811, 304250263527211, 13082761331670031, 614889782588491411, 32589158477190044731, 1922760350154212639071, 117288381359406970983271, 7858321551080267055879091
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OFFSET
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0,1
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COMMENTS
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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
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
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REFERENCES
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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.
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LINKS
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Shubhankar Paul, Ten Problems of Number Theory, International Journal of Engineering and Technical Research (IJETR), ISSN: 2321-0869, Volume-1, Issue-9, November 2013.
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FORMULA
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EXAMPLE
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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
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MAPLE
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with(numtheory): A006862 := proc(n) local i; if n=0 then 2 else 1+product('ithprime(i)', 'i'=1..n); fi; end;
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 2,
1+ithprime(n)*(a(n-1)-1))
end:
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MATHEMATICA
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Table[Product[Prime[k], {k, 1, n}] + 1, {n, 1, 18}]
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PROG
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(Magma) [2] cat [&*PrimesUpTo(p)+1: p in PrimesUpTo(70)]; // Vincenzo Librandi, Dec 03 2015
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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STATUS
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approved
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