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 A034386 Primorial numbers (second definition): n# = product of primes <= n. 247
 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 (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 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. 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 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 # 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 *) 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 (SageMath) 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 (Magma) [n eq 0 select 1 else LCM(PrimesInInterval(1, n)) : n in [0..50]]; // G. C. Greubel, Jul 21 2023 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. Sequence in context: A147299 A090549 A080326 * A083907 A084343 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 7 22:54 EDT 2024. Contains 375018 sequences. (Running on oeis4.)