0,3

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

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.

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.

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

a(5) = a(6) = 2*3*5 = 30;

a(7) = 2*3*5*7 = 210.

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

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 *)

(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

(Sage) def sharp_primorial(n): return sloane.A002110(prime_pi(n))

[sharp_primorial(n) for n in (0..30)] # Giuseppe Coppoletta, Jan 26 2015

(Python)

from sympy import primorial

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

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.

Sequence in context: A147299 A090549 A080326 * A083907 A084343 A025552

Adjacent sequences:  A034383 A034384 A034385 * A034387 A034388 A034389

nonn,easy,nice

N. J. A. Sloane

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

approved