A219107 Number of compositions (ordered partitions) of n into distinct prime parts.

1, 0, 1, 1, 0, 3, 0, 3, 2, 2, 8, 1, 8, 3, 8, 8, 10, 25, 16, 9, 16, 38, 16, 61, 18, 62, 46, 66, 160, 91, 138, 99, 70, 122, 306, 126, 314, 151, 362, 278, 588, 901, 602, 303, 654, 1142, 888, 1759, 892, 1226, 950, 2160, 1230, 3379, 1444, 2372, 2100, 4644, 7416

Offset

0,6

Comments

a(0) = 0 iff n in {1,4,6}.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Formula

a(n) = Sum_{k=0..A024936(n)} A219180(n,k)*k!.

Example

a(5) = 3: [2,3], [3,2], [5].
a(7) = 3: [2,5], [5,2], [7].
a(8) = 2: [3,5], [5,3].
a(9) = 2: [2,7], [7,2].
a(10) = 8: [2,3,5], [2,5,3], [3,2,5], [3,5,2], [5,2,3], [5,3,2], [3,7], [7,3].

Maple

with(numtheory):
b:= proc(n, i) b(n, i):=
      `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));
      a(n):= add(l[i]*(i-1)!, i=1..nops(l))
    end:
seq(a(n), n=0..70);
# second Maple program:
s:= proc(n) option remember; `if`(n<1, 0, ithprime(n)+s(n-1)) end:
b:= proc(n, i, t) option remember; `if`(s(i)<n, 0, `if`(n=0, t!, (p
      ->`if`(p>n, 0, b(n-p, i-1, t+1)))(ithprime(i))+b(n, i-1, t)))
    end:
a:= n-> b(n, numtheory[pi](n), 0):
seq(a(n), n=0..70);  # Alois P. Heinz, Jan 30 2020

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, {}, b[n, i-1] ~zip~ Join[{0}, If[Prime[i] > n, {}, b[n - Prime[i], i-1]]], {0}]];
a[n_] := Module[{l}, l = b[n, PrimePi[n]]; Sum[l[[i]]*(i-1)!, {i, 1, Length[l]}]];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Mar 24 2017, adapted from Maple *)

Crossrefs

Cf. A000142, A000586, A032021, A218396.

Sequence in context: A070298 A024938 A332715 * A338498 A221166 A004604

Adjacent sequences: A219104 A219105 A219106 * A219108 A219109 A219110

Keyword

nonn

Author

Alois P. Heinz, Nov 11 2012

Status

approved