A328946 Product of primorials of consecutive integers (second definition A034386).

1, 1, 2, 12, 72, 2160, 64800, 13608000, 2857680000, 600112800000, 126023688000000, 291114719280000000, 672475001536800000000, 20194424296150104000000000, 606438561613387623120000000000, 18211350005250030322293600000000000, 546886840657658410578476808000000000000

Offset: 0

Author

David S. Metzler, Oct 31 2019

Keywords

• nonn

Comments

Similar to superprimorials (A006939), but a term of the sequence is a product of primorials of consecutive integers, not consecutive primes. So after 2# each primorial will repeat at least twice in the product. Also similar to superprimorials in that the exponents of the primes decrease linearly, but here it is linearly in p, not in pi(p).

Examples

			a(7) = 1# * 2# * 3# * 4# * 5# * 6# * 7# = 1*2*(2*3)*(2*3)*(2*3*5)*(2*3*5)*(2*3*5*7) = 2^6 * 3^5 * 5^3 * 7^1. Note that in the prime factorization the sum of each prime and its exponent is constant and equal to 7+1 = 8.
a(23) = 2^22 * 3^21 * 5^19 * 7^17 * 11^13 * 13^11 * 17^7 * 19^5 * 23^1. Here each prime and its exponent add to 24.
		

Crossrefs

Product of consecutive elements of A034386.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, [1$2], (p-> (h->
      [h, h*p[2]])(`if`(isprime(n), n, 1)*p[1]))(b(n-1)))
    end:
a:= n-> b(n)[2]:
seq(a(n), n=0..16);  # Alois P. Heinz, Nov 11 2020

Mathematica

b[n_] := b[n] = If[n==0, {1, 1}, Function[p, Function[h, {h, h p[[2]]}][If[ PrimeQ[n], n, 1] p[[1]]]][b[n - 1]]];
a[n_] := b[n][[2]];
a /@ Range[0, 16] (* Jean-François Alcover, Nov 30 2020, after Alois P. Heinz *)

PARI

primo(n) = lcm(primes([2, n])); \\ A034386
a(n) = prod(k=1, n, primo(k)); \\ Michel Marcus, Nov 01 2019

Formula

a(n) = Product_{k=1..n} A034386(k) = Product_{p prime, p<=n} p^(n-p+1) = Product_{p prime} p^(max(n-p+1,0)) = Product_{p prime,p+k = n+1 and k >= 0} p^k.

a(n) = lcm(n, a(n-1)^2/a(n-2)). - Jon Maiga, Jul 08 2021

Extensions

a(0)=1 prepended by Alois P. Heinz, Nov 11 2020