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

0,6

Links

Table of n, a(n) for n=0..55.

Formula

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

Example

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

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 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

Crossrefs

Cf. A240850, A000009.

Sequence in context: A349464 A325260 A325325 * A261665 A208096 A049269

Adjacent sequences: A240848 A240849 A240850 * A240852 A240853 A240854

Keyword

nonn,easy

Author

Clark Kimberling, Apr 14 2014

Status

approved