%I #25 Sep 12 2024 14:45:02

%S 1,1,2,3,2,10,6,14,6,6,30,66,30,78,42,30,30,510,210,570,210,210,330,

%T 690,2310,210,2730,210,2310,6090,30030,6510,2730,2310,39270,2310,

%U 46410,85470,3990,30030,39270,94710,570570,1291290,30030,30030,903210,1411410,746130

%N Product of all distinct least part primes from all partitions of n into prime parts.

%C For all n, omega(a(n)) = Omega(a(n)). The prime factorization of each term gives the least part primes of all partitions of n into prime parts.

%C Product of all terms in row n of A333238. - _Alois P. Heinz_, Mar 16 2020

%H Alois P. Heinz, <a href="/A333129/b333129.txt">Table of n, a(n) for n = 0..6269</a>

%e a(2) = 2 because [2] is the only prime partition of 2. a(5) = 10 because the prime partitions of 5 are [2,3] and [5], so the products of all distinct least part primes is 2*5 = 10.

%p b:= proc(n, p, t) option remember; `if`(n=0, 1, `if`(p>n, 0, (q->

%p add(b(n-p*j, q, 1), j=1..n/p)*t^p+b(n, q, t))(nextprime(p))))

%p end:

%p a:= n-> (p-> mul(`if`(coeff(p, x, i)>0, i, 1), i=2..n))(b(n, 2, x)):

%p seq(a(n), n=0..55); # _Alois P. Heinz_, Mar 12 2020

%t a[0] = 1; a[n_] := Times @@ Union[Min /@ IntegerPartitions[n, All, Prime[ Range[PrimePi[n]]]]];

%t a /@ Range[0, 55] (* _Jean-François Alcover_, Nov 01 2020 *)

%Y Cf. A000040, A000607, A001221, A001222, A005117, A333238.

%K nonn

%O 0,3

%A _David James Sycamore_, Mar 08 2020