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A240851
Number of partitions p of n into distinct parts not including mean(p).
10
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
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
Sequence in context: A349464 A325260 A325325 * A261665 A208096 A049269
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 14 2014
STATUS
approved