A034386 Primorial numbers (second definition): n# = product of primes <= n.

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, 200560490130, 200560490130

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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(n-1) and taking a(n) = a(n-1) otherwise. - Amarnath Murthy, Oct 30 2002; corrected by Franklin T. Adams-Watters, Dec 13 2006

Examples

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

References

• Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3, p. 14, "n?".
• József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Section VII.35, p. 268.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2370 (first 401 terms from T. D. Noe)
• Jens Askgaard, On the additive period length of the Sprague-Grundy function of certain Nim-like games, arXiv:1902.06299 [math.CO], 2019.
• Klaus Dohmen and Martin Trinks, An Abstraction of Whitney's Broken Circuit Theorem, arXiv:1404.5480 [math.CO], 2014.
• Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv:1202.3670 [math.HO], 2012. - From _N. J. A. Sloane_, Jun 13 2012
• J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math., Vol. 6, No. 1 (1962), 64-94.
• Eric Weisstein's World of Mathematics, Primorial.

Crossrefs

Cf. A002110, A057872, A249270.

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.

Programs

Magma

[n eq 0 select 1 else LCM(PrimesInInterval(1, n)) : n in [0..50]]; // G. C. Greubel, Jul 21 2023

Maple

A034386 := n -> mul(k,k=select(isprime,[$1..n])); # Peter Luschny, Jun 19 2009
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1,
      `if`(isprime(n), n, 1)*a(n-1))
    end:
seq(a(n), n=0..36);  # Alois P. Heinz, Nov 26 2020

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 *)
Table[ResourceFunction["Primorial"][i], {i,1,40}] (* Navvye Anand, May 22 2024 *)

PARI

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

PARI

a(n)=lcm(primes([2,n])) \\ Jeppe Stig Nielsen, Mar 10 2019

Python

from sympy import primorial
def A034386(n): return 1 if n == 0 else primorial(n,nth=False) # Chai Wah Wu, Jan 11 2022

SageMath

def sharp_primorial(n): return sloane.A002110(prime_pi(n))
[sharp_primorial(n) for n in (0..30)] # Giuseppe Coppoletta, Jan 26 2015

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

For n > 0, log(a(n)) < 1.01624*n. [Rosser and Schoenfeld, 1962; Sándor et al., 2005] - N. J. A. Sloane, Apr 04 2017

a(n) <= A179215(n). - Reinhard Zumkeller, Jul 05 2010

a(n) = lcm(A006530(n), a(n-1)). - Jon Maiga, Nov 10 2018

Sum_{n>=0} 1/a(n) = A249270. - Amiram Eldar, Nov 08 2020

Extensions

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