

A034386


Primorial numbers (second definition): n# = product of primes <= n.


112



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(n1) and taking a(n) = a(n1) otherwise.  Amarnath Murthy, Oct 30 2002; corrected by Franklin T. AdamsWatters, Dec 13 2006
If n = a(n1) + 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).  Reinhard Zumkeller, Jul 05 2010


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 [math.CO], 2014.
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 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


EXAMPLE

a(5) = a(6) = 2*3*5 = 30
a(7) = 2*3*5*7 = 210


MAPLE

A034386 := n > mul(k, k=select(isprime, [$1..n])); # 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}]] (* 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
(Sage) def sharp_primorial(n): return sloane.A002110(prime_pi(n))
[sharp_primorial(n) for n in (0..30)] # Giuseppe Coppoletta, Jan 26 2015


CROSSREFS

Cf. A002110, A057872.
Cf. A073838, A034387.  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,changed


AUTHOR

N. J. A. Sloane


EXTENSIONS

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


STATUS

approved



