A393675 a(n) = number of partitions of n into distinct composite parts, no two of which are consecutive in A002808.

1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 2, 2, 3, 0, 4, 1, 4, 3, 5, 2, 7, 5, 6, 7, 9, 6, 12, 8, 12, 11, 16, 10, 22, 14, 20, 20, 28, 20, 33, 27, 34, 36, 43, 36, 56, 48, 58, 58, 71, 62, 88, 76, 92, 93, 114, 98, 139, 122, 148, 149, 178, 161, 209, 195, 228, 233

Offset

0,13

Links

Table of n, a(n) for n=0..69.

Example

a(16) counts these 3 partitions: 16, 12+4, 10+6.

Mathematica

prt[n_, sL_] := Module[{m = Length[sL], f}, If[n > Last[sL], Return["Extend sL"]];
f[i_, r_, b_] := f[i, r, b] = Which[r == 0, {{}}, r < 0 || i == 0, {}, True,
Join[f[i - 1, r, False], If[! b && s[[i]] <= r, Map[Prepend[#, sL[[i]]] &,
f[i - 1, r - sL[[i]], True]], {}]]];  Reverse[f[m, n, False]]]
z = 100;
s = Select[Range[z], CompositeQ];
m = Map[{#, prt[#, s]} &, Range[z]]
u = Map[Last, m];
Map[Length, u]
(* Peter J. C. Moses, Jan 26 2026 *)

Crossrefs

Cf. A018252, A393671, A393672, A393673, A393674, A393676.

Sequence in context: A071445 A144081 A278838 * A140086 A037194 A326130

Adjacent sequences: A393672 A393673 A393674 * A393676 A393677 A393678

Keyword

nonn

Author

Clark Kimberling, Mar 10 2026

Status

approved