login
Number of partitions of n whose mean is a part.
90

%I #12 Sep 22 2023 01:59:00

%S 1,2,2,3,2,5,2,6,5,8,2,21,2,14,22,30,2,61,2,86,67,45,2,283,66,80,197,

%T 340,2,766,2,663,543,234,703,2532,2,388,1395,4029,2,4688,2,4476,7032,

%U 1005,2,17883,2434,9713,7684,14472,2,25348,17562,37829,16786,3721

%N Number of partitions of n whose mean is a part.

%C a(n) = 2 if and only if n is a prime.

%F a(n) = A000041(n) - A327472(n). - _Gus Wiseman_, Sep 14 2019

%e a(6) counts these partitions: 6, 33, 321, 222, 111111.

%e From _Gus Wiseman_, Sep 14 2019: (Start)

%e The a(1) = 1 through a(10) = 8 partitions (A = 10):

%e 1 2 3 4 5 6 7 8 9 A

%e 11 111 22 11111 33 1111111 44 333 55

%e 1111 222 2222 432 22222

%e 321 3221 531 32221

%e 111111 4211 111111111 33211

%e 11111111 42211

%e 52111

%e 1111111111

%e (End)

%t Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Mean[p]]], {n, 40}]

%o (Python)

%o from sympy.utilities.iterables import partitions

%o def A237984(n): return sum(1 for s,p in partitions(n,size=True) if not n%s and n//s in p) # _Chai Wah Wu_, Sep 21 2023

%Y Cf. A238478.

%Y The Heinz numbers of these partitions are A327473.

%Y A similar sequence for subsets is A065795.

%Y Dominated by A067538.

%Y The strict case is A240850.

%Y Partitions without their mean are A327472.

%Y Cf. A000016, A316413, A324753, A325705, A327478, A327482.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Feb 27 2014