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

0,17

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}.

Links

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

Formula

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

Example

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.

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 *)

Crossrefs

Cf. A219180.

Sequence in context: A271721 A050252 A025877 * A184171 A362716 A133989

Adjacent sequences: A219179 A219180 A219181 * A219183 A219184 A219185

Keyword

nonn

Author

Alois P. Heinz, Nov 13 2012

Status

approved