A240850 Number of partitions p of n into distinct parts including mean(p).

0, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 5, 1, 1, 6, 5, 1, 6, 1, 14, 7, 1, 1, 24, 16, 1, 9, 23, 1, 58, 1, 31, 11, 1, 75, 103, 1, 1, 13, 163, 1, 202, 1, 66, 182, 1, 1, 413, 203, 246, 17, 97, 1, 550, 347, 889, 19, 1, 1, 1500, 1, 1, 1442, 982, 625, 1424, 1, 177, 23

Offset

0,7

Links

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

Formula

a(n) + A240851(n) = A000009(n) for n >= 0.

Example

a(12) counts these 5 partitions:  {12}, {7,4,1}, {6,4,2}, {6,3,2,1}, {5,4,3}.

Mathematica

z = 70; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
Table[Count[f[n], p_ /; MemberQ[p, Mean[p]]], {n, 0, z}]   (* A240850 *)
Table[Count[f[n], p_ /; ! MemberQ[p, Mean[p]]], {n, 0, z}] (* A240851 *)

Prog

(Python)
from sympy.utilities.iterables import partitions
def A240850(n): return sum(1 for s, p in partitions(n, size=True) if max(p.values(), default=0)==1 and not n%s and n//s in p) # Chai Wah Wu, Sep 21 2023

Crossrefs

Cf. A240851, A000009.

Sequence in context: A307828 A280698 A217667 * A357943 A194086 A342723

Adjacent sequences: A240847 A240848 A240849 * A240851 A240852 A240853

Keyword

nonn,easy

Author

Clark Kimberling, Apr 14 2014

Status

approved