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

0,3

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).

Links

Table of n, a(n) for n=0..16.

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

Example

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.

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 *)

Prog

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

Crossrefs

Product of consecutive elements of A034386.

Sequence in context: A130426 A002397 A163085 * A037515 A037718 A342054

Adjacent sequences: A328943 A328944 A328945 * A328947 A328948 A328949

Keyword

nonn

Author

David S. Metzler, Oct 31 2019

Extensions

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

Status

approved