A219181 Number of partitions of n into the maximal possible number 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, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 1, 2, 1, 3, 1, 3, 2, 4, 2, 4, 2, 5, 2, 1, 4, 1, 4, 1, 4, 1, 6, 2, 6, 1, 6, 2, 8, 4, 10, 2, 1, 5, 1, 6, 1, 5, 2, 6, 2, 10, 1, 9, 1, 11, 4, 15, 3, 14, 3, 1, 6, 1, 6, 1, 5, 1, 10, 1, 11

Offset

0,19

Comments

a(n) is the last 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) = A219180(n,A024936(n)) if A024936(n) >= 0, a(n) = 0 else.

Example

a(18) = 2 because there are 2 partitions of 18 into 3 distinct prime parts ([2,3,13], [2,5,11]) but no partitions of 18 into more than 3 distinct prime parts.

Maple

with(numtheory):
b:= proc(n, i) option remember;
      `if`(n=0, [1], `if`(i<1, [], zip((x, y)->x+y, b(n, i-1),
       [0, `if`(ithprime(i)>n, [], b(n-ithprime(i), i-1))[]], 0)))
    end:
a:= proc(n) local l; l:=b(n, pi(n));
       while nops(l)>0 and l[-1]=0 do
          l:= subsop(-1=NULL, l)
       od;
       `if`(nops(l)=0, 0, l[-1])
    end:
seq(a(n), n=0..100);

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, {}, zip[b[n, i-1], Join[{0}, If[Prime[i]>n, {}, b[n-Prime[i], i-1]]]]]]; a[n_] := (l = b[n, PrimePi[n]]; While[Length[l]>0 && l[[-1]] == 0, l = ReplacePart[l, -1 -> Nothing]]; If[Length[l] == 0, 0, l[[-1]]]);  Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 12 2017, translated from Maple *)

Crossrefs

Sequence in context: A191613 A298642 A243404 * A347833 A260341 A109969

Adjacent sequences: A219178 A219179 A219180 * A219182 A219183 A219184

Keyword

nonn,look

Author

Alois P. Heinz, Nov 13 2012

Status

approved