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

Author

Clark Kimberling, Apr 14 2014

Keywords

• nonn
• easy

Examples

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

Crossrefs

Cf. A240851, A000009.

Programs

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 *)

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

Formula

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