A219182 - OEIS

%I #13 Feb 12 2017 12:16:41

%S 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,

%T 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,

%U 13,18,14,19,15,21,15,22,18,25,18,26,22,29,22

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

%C 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}.

%H Alois P. Heinz, <a href="/A219182/b219182.txt">Table of n, a(n) for n = 0..3500</a>

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

%e 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.

%p with(numtheory):

%p b:= proc(n, i) option remember;

%p       `if`(n=0, [1], `if`(i<1, [0], zip((x, y)->x+y, b(n, i-1),

%p        [0, `if`(ithprime(i)>n, [], b(n-ithprime(i), i-1))[]], 0)))

%p     end:

%p a:= n-> max(b(n, pi(n))[]):

%p seq(a(n), n=0..120);

%t 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 *)

%Y Cf. A219180.

%K nonn

%O 0,17

%A _Alois P. Heinz_, Nov 13 2012