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

1, 1, 2, 3, 2, 10, 6, 14, 6, 6, 30, 66, 30, 78, 42, 30, 30, 510, 210, 570, 210, 210, 330, 690, 2310, 210, 2730, 210, 2310, 6090, 30030, 6510, 2730, 2310, 39270, 2310, 46410, 85470, 3990, 30030, 39270, 94710, 570570, 1291290, 30030, 30030, 903210, 1411410, 746130

Offset: 0

Author

David James Sycamore, Mar 08 2020

Keywords

• nonn

Comments

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.

Product of all terms in row n of A333238. - Alois P. Heinz, Mar 16 2020

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..6269

Crossrefs

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

Programs

Maple

b:= proc(n, p, t) option remember; `if`(n=0, 1, `if`(p>n, 0, (q->
      add(b(n-p*j, q, 1), j=1..n/p)*t^p+b(n, q, t))(nextprime(p))))
    end:
a:= n-> (p-> mul(`if`(coeff(p, x, i)>0, i, 1), i=2..n))(b(n, 2, x)):
seq(a(n), n=0..55);  # Alois P. Heinz, Mar 12 2020

Mathematica

a[0] = 1; a[n_] := Times @@ Union[Min /@ IntegerPartitions[n, All, Prime[ Range[PrimePi[n]]]]];
a /@ Range[0, 55] (* Jean-François Alcover, Nov 01 2020 *)