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Numbers for which there exists an integer partition such that the parts have the same mean as the multiplicities.
1

%I #18 Jan 29 2023 21:02:11

%S 1,4,8,9,12,16,18,20,25,27,32,36,45,48,49,50,54,63,64,72,75,80,81,90,

%T 96,98,99,100,108,112,117,121,125,128,144,147,150,160,162,169,175,176,

%U 180,192,196,200,208,216,224,225,240,242,243,245,250,252,256,272

%N Numbers for which there exists an integer partition such that the parts have the same mean as the multiplicities.

%C Conjecture: No term > 1 is squarefree.

%e A partition of 20 with the same mean as its multiplicities is (5,4,3,2,1,1,1,1,1,1), so 20 is in the sequence.

%t Select[Range[30],Select[IntegerPartitions[#],Mean[#]==Mean[Length/@Split[#]]&]!={}&]

%Y Positions of positive terms in A360068, ranked by A359903.

%Y A000041 counts partitions, strict A000009.

%Y A058398 counts partitions by mean, see also A008284, A327482.

%Y A088529/A088530 gives mean of prime signature (A124010).

%Y A326567/A326568 gives mean of prime indices (A112798).

%Y Cf. A005117, A067340, A067538, A240219, A316313, A349156, A359904, A359905, A360069.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jan 27 2023

%E a(22)-a(58) from _Alois P. Heinz_, Jan 29 2023