A194261 Smallest prime that divides the n-th partition number p(n) but does not divide p(1)*p(2)*...*p(n-1), or 1 if none.

1, 2, 3, 5, 7, 11, 1, 1, 1, 1, 1, 1, 101, 1, 1, 1, 1, 1, 1, 19, 1, 167, 251, 1, 89, 29, 43, 13, 83, 467, 311, 23, 1, 1231, 41, 17977, 281, 1, 1, 127, 193, 2417, 71, 31, 1087, 73, 67, 7013, 631, 9283, 661, 53, 5237, 17, 227, 47, 102359, 3251, 199, 139, 971, 2273

Offset: 1

Author

Jonathan Sondow, Aug 20 2011

Keywords

• nonn
• look

Comments

It appears that a(n) is prime for all n > 97. See A194259 and A194260 for additional comments and links.

Links

• Alois P. Heinz and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 2000 terms from Alois P. Heinz)

Crossrefs

Cf. A000041, A194259, A194260, A194262.

Programs

Maple

with(combinat): with(numtheory):
b:= proc(n) option remember;
      `if`(n=1, {}, b(n-1) union factorset(numbpart(n)))
    end:
m:= proc(n) option remember; min((b(n) minus b(n-1))[]) end:
a:= n-> `if`(n=1, 1, `if`(m(n)=infinity, 1, m(n))):
seq(a(n), n=1..120);  # Alois P. Heinz, Aug 21 2011

Mathematica

a[n_] := Complement[FactorInteger[PartitionsP[n]][[All, 1]], FactorInteger[Product[PartitionsP[k], {k, 1, n-1}]][[All, 1]]] /. {} -> {1} // First; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 28 2014 *)