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A034386 Primorial numbers (second definition): n# = product of primes <= n. 234
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)
Klaus Dohmen and Martin Trinks, An Abstraction of Whitney's Broken Circuit Theorem, arXiv:1404.5480 [math.CO], 2014.
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 *)
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. 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
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 March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)