

A034386


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


146



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

0,3


COMMENTS

Squarefree kernel of both n! and lcm(1, 2, 3, ..., 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
For sufficiently large n, a(n) = 2n  2 according to the link below by Ballar, et al., see also the Apr 11, 2017 Quanta Magazine article.  W. Edwin Clark, Apr 11 2017


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
I. Ballar, F. Draxler, P. Keevash, B. Sudakov, Equiangular Lines and Spherical Codes in Euclidean Space, arxiv preprint arxiv:1606.06620 [math.HO], 2016.
K. Dohmen, M. Trinks, An Abstraction of Whitney's Broken Circuit Theorem, arXiv preprint arXiv:1404.5480 [math.CO], 2014.
K. Hartnett, A New Path To Equal Angle Lines, Quanta Magazine, Apr 11, 2017.
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

For n > 0, log(a(n)) < 1.01624*n. [Rosser and Schoenfeld, Ill. J. Math. (1962); Mitrinovic, H'book of Number Theory, Section VII.35, p. 268.]  N. J. A. Sloane, Apr 04 2017
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
a(n) = lcm(A006530(n), a(n1)).  Jon Maiga, Nov 10 2018


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
The following fractions are all related to each other: Sum 1/n: A001008/A002805, Sum 1/prime(n): A024451/A002110 and A106830/A034386, Sum 1/nonprime(n): A282511/A282512, Sum 1/composite(n): A250133/A296358.
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



