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

0,3

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

Links

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A126288 A205393 A075076 * A078828 A247170 A241055

Adjacent sequences: A333126 A333127 A333128 * A333130 A333131 A333132

Keyword

nonn

Author

David James Sycamore, Mar 08 2020

Status

approved