%I #7 Jun 12 2023 08:44:19

%S 1,32,40,60,100,126,210,243,294,351,550,585,770,819,1210,1274,1275,

%T 1287,1521,1785,2002,2366,2793,2805,2875,3125,3315,4025,4114,4335,

%U 4389,4862,5187,6325,6358,6422,6783,7105,7475,7581,8349,8398,9386,9775,9867,10925

%N Heinz numbers of integer partitions such that 3*(sum) = (weighted sum).

%C The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

%C The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. For example, the weighted sum of (4,2,2,1) is 1*4 + 2*2 + 3*2 + 4*1 = 18.

%F A056239(a(n)) = A304818(a(n))/3.

%e The terms together with their prime indices begin:

%e 1: {}

%e 32: {1,1,1,1,1}

%e 40: {1,1,1,3}

%e 60: {1,1,2,3}

%e 100: {1,1,3,3}

%e 126: {1,2,2,4}

%e 210: {1,2,3,4}

%e 243: {2,2,2,2,2}

%e 294: {1,2,4,4}

%e 351: {2,2,2,6}

%e 550: {1,3,3,5}

%e 585: {2,2,3,6}

%e 770: {1,3,4,5}

%e 819: {2,2,4,6}

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[1000],3*Total[prix[#]]==Total[Accumulate[Reverse[prix[#]]]]&]

%Y These partitions are counted by A363527.

%Y The reverse version is A363531, counted by A363526.

%Y A053632 counts compositions by weighted sum.

%Y A055396 gives minimum prime index, maximum A061395.

%Y A112798 lists prime indices, length A001222, sum A056239.

%Y A264034 counts partitions by weighted sum, reverse A358194.

%Y A304818 gives weighted sum of prime indices, row-sums of A359361.

%Y A318283 gives weighted sum of reversed prime indices, row-sums of A358136.

%Y A320387 counts multisets by weighted sum, zero-based A359678.

%Y Cf. A000041, A000720, A001221, A046660, A106529, A118914, A124010, A181819, A215366, A359362, A359755.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jun 12 2023