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 A034386 Primorial numbers (second definition): n# = product of primes <= n. 154
 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, 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 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 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 Jens Askgaard, On the additive period length of the Sprague-Grundy function of certain Nim-like games, arXiv:1902.06299 [math.CO], 2019. K. Dohmen, M. Trinks, An Abstraction of Whitney's Broken Circuit Theorem, arXiv:1404.5480 [math.CO], 2014. R. Mestrovic, 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 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(n-1)). - 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 (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 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 EXTENSIONS Offset changed and initial term added by Arkadiusz Wesolowski, Jun 04 2011 STATUS approved

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Last modified October 20 02:03 EDT 2019. Contains 328244 sequences. (Running on oeis4.)