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A034386
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Primorial numbers (second definition): n# = product of primes <= n.
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95
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1, 1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, 2310, 30030, 30030, 30030, 30030, 510510, 510510, 9699690, 9699690, 9699690, 9699690, 223092870, 223092870, 223092870, 223092870, 223092870, 223092870, 6469693230, 6469693230
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OFFSET
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0,3
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COMMENTS
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Squarefree kernel of both n! and lcm{1..n}.
a(n)=lcm{ core(1),core(2),core(3),...,core(n)} where core(x) denotes the squarefree part of x, the smallest integer such that x*core(x) is a square. - Benoit Cloitre, May 31 2002
The sequence can also be obtained by taking a(1) = 1 and then multiplying the previous term by n if n is coprime to the previous term a(n-1) and taking a(n) = a(n-1) otherwise. - Amarnath Murthy, Oct 30 2002; corrected by Franklin T. Adams-Watters, Dec 13 2006
If n = a(n-1) + 1, then n is prime. However, this is only satisfied for trivial cases n=2 and n=3. - Matthew Flaschen (matthew.flaschen(AT)gatech.edu), May 24 2008
a(n) <= A179215(n). [From Reinhard Zumkeller, Jul 05 2010]
a(0) is also defined. It has the value of the empty product, hence 1. - Peter Luschny, Mar 05 2011.
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REFERENCES
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S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3, p. 14, "n?".
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
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..400
Eric Weisstein's World of Mathematics, Primorial
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FORMULA
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a(n) = n# = A002110(A000720(n)) = A007947(A003418(n)) = A007947(A000142(n)).
Asymptotic expression for a(n): exp((1 + o(1)) * n) where o(1) is the "little o" notation - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 08 2001
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MAPLE
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A034386 := n -> mul(k, k=select(isprime, [$1..n])); [From Peter Luschny, Jun 19 2009]
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MATHEMATICA
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q[x_]:=Apply[Times, Table[Prime[w], {w, 1, PrimePi[x]}]]; Table[q[w], {w, 1, 30}]
With[{pr=FoldList[Times, 1, Prime[Range[20]]]}, Table[pr[[PrimePi[n]+1]], {n, 0, 40}]] (* From Harvey P. Dale, Apr 05 2012 *)
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PROG
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(PARI) a(n)=my(v=primes(primepi(n))); prod(i=1, #v, v[i]) \\ Charles R Greathouse IV, Jun 15 2011
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CROSSREFS
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Cf. A002110, A057872.
Cf. A073838, A034387. [From Reinhard Zumkeller, Jul 05 2010]
Sequence in context: A147299 A090549 A080326 * A084343 A083907 A025552
Adjacent sequences: A034383 A034384 A034385 * A034387 A034388 A034389
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Offset changed and initial term added by Arkadiusz Wesolowski, Jun 04 2011
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STATUS
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approved
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