A240851 Number of partitions p of n into distinct parts not including mean(p).

0, 0, 0, 1, 1, 2, 2, 4, 5, 5, 9, 11, 10, 17, 21, 21, 27, 37, 40, 53, 50, 69, 88, 103, 98, 126, 164, 183, 199, 255, 238, 339, 359, 437, 511, 510, 565, 759, 863, 969, 950, 1259, 1224, 1609, 1750, 1866, 2303, 2589, 2497, 3061, 3412, 4080, 4485, 5119, 5168, 6031

Offset: 0

Author

Clark Kimberling, Apr 14 2014

Keywords

• nonn
• easy

Examples

			a(9) counts these 5 partitions:  81, 72, 63. 621, 54.
		

Crossrefs

Cf. A240850, 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 A240851(n): return sum(1 for s,p in partitions(n,size=True) if max(p.values(),default=0)==1 and (n%s or n//s not in p)) # Chai Wah Wu, Sep 21 2023

Formula

a(n) + A240850(n) = A000009(n) for n >= 0.