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

%I A328946 #25 Jul 09 2021 05:24:20
%S A328946 1,1,2,12,72,2160,64800,13608000,2857680000,600112800000,
%T A328946 126023688000000,291114719280000000,672475001536800000000,
%U A328946 20194424296150104000000000,606438561613387623120000000000,18211350005250030322293600000000000,546886840657658410578476808000000000000
%N A328946 Product of primorials of consecutive integers (second definition A034386).
%C A328946 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).
%F A328946 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.
%F A328946 a(n) = lcm(n, a(n-1)^2/a(n-2)). - _Jon Maiga_, Jul 08 2021
%e A328946 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.
%e A328946 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.
%p A328946 b:= proc(n) option remember; `if`(n=0, [1$2], (p-> (h->
%p A328946       [h, h*p[2]])(`if`(isprime(n), n, 1)*p[1]))(b(n-1)))
%p A328946     end:
%p A328946 a:= n-> b(n)[2]:
%p A328946 seq(a(n), n=0..16);  # _Alois P. Heinz_, Nov 11 2020
%t A328946 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]]];
%t A328946 a[n_] := b[n][[2]];
%t A328946 a /@ Range[0, 16] (* _Jean-François Alcover_, Nov 30 2020, after _Alois P. Heinz_ *)
%o A328946 (PARI) primo(n) = lcm(primes([2, n])); \\ A034386
%o A328946 a(n) = prod(k=1, n, primo(k)); \\ _Michel Marcus_, Nov 01 2019
%Y A328946 Product of consecutive elements of A034386.
%K A328946 nonn
%O A328946 0,3
%A A328946 _David S. Metzler_, Oct 31 2019
%E A328946 a(0)=1 prepended by _Alois P. Heinz_, Nov 11 2020