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A333129 Product of all distinct least part primes from all partitions of n into prime parts.

%I A333129 #25 Sep 12 2024 14:45:02
%S A333129 1,1,2,3,2,10,6,14,6,6,30,66,30,78,42,30,30,510,210,570,210,210,330,
%T A333129 690,2310,210,2730,210,2310,6090,30030,6510,2730,2310,39270,2310,
%U A333129 46410,85470,3990,30030,39270,94710,570570,1291290,30030,30030,903210,1411410,746130
%N A333129 Product of all distinct least part primes from all partitions of n into prime parts.
%C A333129 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 A333129 Product of all terms in row n of A333238. - _Alois P. Heinz_, Mar 16 2020
%H A333129 Alois P. Heinz, <a href="/A333129/b333129.txt">Table of n, a(n) for n = 0..6269</a>
%e A333129 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 A333129 b:= proc(n, p, t) option remember; `if`(n=0, 1, `if`(p>n, 0, (q->
%p A333129       add(b(n-p*j, q, 1), j=1..n/p)*t^p+b(n, q, t))(nextprime(p))))
%p A333129     end:
%p A333129 a:= n-> (p-> mul(`if`(coeff(p, x, i)>0, i, 1), i=2..n))(b(n, 2, x)):
%p A333129 seq(a(n), n=0..55);  # _Alois P. Heinz_, Mar 12 2020
%t A333129 a[0] = 1; a[n_] := Times @@ Union[Min /@ IntegerPartitions[n, All, Prime[ Range[PrimePi[n]]]]];
%t A333129 a /@ Range[0, 55] (* _Jean-François Alcover_, Nov 01 2020 *)
%Y A333129 Cf. A000040, A000607, A001221, A001222, A005117, A333238.
%K A333129 nonn
%O A333129 0,3
%A A333129 _David James Sycamore_, Mar 08 2020