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A034386
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Primorial numbers (second definition): n# = product of primes <= n.
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231
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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
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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Eric Weisstein's World of Mathematics, Primorial.
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FORMULA
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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
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EXAMPLE
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a(5) = a(6) = 2*3*5 = 30;
a(7) = 2*3*5*7 = 210.
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MAPLE
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# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1,
`if`(isprime(n), n, 1)*a(n-1))
end:
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MATHEMATICA
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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 *)
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PROG
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(Sage) def sharp_primorial(n): return sloane.A002110(prime_pi(n))
(Python)
from sympy import primorial
def A034386(n): return 1 if n == 0 else primorial(n, nth=False) # Chai Wah Wu, Jan 11 2022
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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