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A034386 Primorial numbers (second definition): n# = product of primes <= n. 124
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). - 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<\a>, 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<\a>, Quanta Magazine, Apr 11, 2017.

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 [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 Schoenfield, 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

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

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 19 07:12 EDT 2017. Contains 290794 sequences.