A219182 Maximal number of partitions of n into any number k of distinct prime parts or 0 if there are no such partitions.

1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 3, 2, 3, 3, 4, 3, 4, 3, 5, 4, 5, 4, 6, 4, 6, 4, 6, 6, 6, 6, 9, 6, 9, 8, 8, 10, 11, 10, 11, 11, 11, 13, 13, 14, 13, 16, 13, 18, 14, 19, 15, 21, 15, 22, 18, 25, 18, 26, 22, 29, 22

Offset: 0

Author

Alois P. Heinz, Nov 13 2012

Keywords

• nonn

Comments

a(n) is maximal element of row n of triangle A219180 or 0 if the row is empty. a(n) = 0 iff n in {1,4,6}.

Examples

			a(31) = 4 because there are 4 partitions of 31 into 3 distinct prime parts ([3,5,23], [3,11,17], [5,7,19], [7,11,13]) but not more than 4 partitions of 31 into k distinct prime parts for any other k.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..3500

Crossrefs

Cf. A219180.

Programs

Maple

with(numtheory):
b:= proc(n, i) option remember;
      `if`(n=0, [1], `if`(i<1, [0], zip((x, y)->x+y, b(n, i-1),
       [0, `if`(ithprime(i)>n, [], b(n-ithprime(i), i-1))[]], 0)))
    end:
a:= n-> max(b(n, pi(n))[]):
seq(a(n), n=0..120);

Mathematica

zip = With[{m = Max[Length[#1], Length[#2]]}, PadRight[#1, m] + PadRight[#2, m]]&; b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i<1, {0}, zip[b[n, i-1], Join[{0}, If[Prime[i]>n, {}, b[n-Prime[i], i-1]]]]]]; a[n_] := Max[b[n, PrimePi[n]]]; Table[a[n], {n, 0, 120}] (* Jean-François Alcover, Feb 12 2017, translated from Maple *)

Formula

a(n) = max_{k>=0} A219180(n,k).