A272899 Product of next n prime numbers greater than n.

1, 2, 15, 385, 5005, 323323, 7436429, 955049953, 35336848261, 1448810778701, 62298863484143, 14107860812636383, 832363787945546597, 261682369333342226303, 18579448222667298067513, 1356299720254712758928449, 107147677900122307955347471, 46558817449894322874479515781

Offset: 0

Author

Matthew Goers, May 09 2016

Keywords

• nonn
• easy

Comments

a(n) is of comparable size to n^n. - Charles R Greathouse IV, May 09 2016

a(n) is the product of the terms of the n-th row of A084754. - Michel Marcus, May 09 2016

Examples

			a(0) = 1 (the empty product).
a(1) = 2 = 2.
a(2) = 3 * 5 = 15.
a(3) = 5 * 7 * 11 = 385.
a(4) = 5 * 7 * 11 * 13 = 5005.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..322

Crossrefs

Cf. A000720, A002110, A007918, A034386, A084754.

Programs

Maple

a:= n-> mul((nextprime@@i)(n), i=1..n):
seq(a(n), n=0..17);  # Alois P. Heinz, Jun 24 2024

Mathematica

Table[Times@@Prime[Range[PrimePi[n] + 1, PrimePi[n] + n]], {n, 25}] (* Alonso del Arte, May 09 2016 *)

PARI

a(n)=my(v=primes(primepi(n)+n)); prod(i=0,n-1,v[#v-i]) \\ Charles R Greathouse IV, May 09 2016

Python

from math import prod
from sympy import prime, primepi
def a(n): r = primepi(n); return prod(prime(i) for i in range(r+1, r+n+1))
print([a(n) for n in range(1, 17)]) # Michael S. Branicky, Feb 15 2021

Formula

a(n) = A002110(n + A000720(n))/A034386(n), where A002110(n) are the primorials, A000720(n) is the pi(n) prime counting function, and A034386(n) is the primorial of primes less than or equal to n. E.g., a(7) = 955049953 = A002110(11) / A034386(7).

Extensions

a(0)=1 prepended by Alois P. Heinz, Jun 24 2024