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A034386 Primorial numbers (second definition): n# = product of primes <= n. 102
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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.

REFERENCES

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3, p. 14, "n?".

LINKS

T. D. Noe, Table of n, a(n) for n = 0..400

K. Dohmen, M. Trinks, An Abstraction of Whitney's Broken Circuit Theorem, arXiv preprint arXiv:1404.5480, 2014

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

Eric Weisstein's World of Mathematics, Primorial

FORMULA

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

MAPLE

A034386 := n -> mul(k, k=select(isprime, [$1..n])); [From Peter Luschny, Jun 19 2009]

MATHEMATICA

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 *)

PROG

(PARI) a(n)=my(v=primes(primepi(n))); prod(i=1, #v, v[i]) \\ Charles R Greathouse IV, Jun 15 2011

CROSSREFS

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

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Offset changed and initial term added by Arkadiusz Wesolowski, Jun 04 2011

STATUS

approved

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Last modified August 27 23:31 EDT 2014. Contains 246149 sequences.