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A363530 Heinz numbers of integer partitions such that 3*(sum) = (weighted sum).

%I A363530 #7 Jun 12 2023 08:44:19
%S A363530 1,32,40,60,100,126,210,243,294,351,550,585,770,819,1210,1274,1275,
%T A363530 1287,1521,1785,2002,2366,2793,2805,2875,3125,3315,4025,4114,4335,
%U A363530 4389,4862,5187,6325,6358,6422,6783,7105,7475,7581,8349,8398,9386,9775,9867,10925
%N A363530 Heinz numbers of integer partitions such that 3*(sum) = (weighted sum).
%C A363530 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 A363530 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 A363530 A056239(a(n)) = A304818(a(n))/3.
%e A363530 The terms together with their prime indices begin:
%e A363530       1: {}
%e A363530      32: {1,1,1,1,1}
%e A363530      40: {1,1,1,3}
%e A363530      60: {1,1,2,3}
%e A363530     100: {1,1,3,3}
%e A363530     126: {1,2,2,4}
%e A363530     210: {1,2,3,4}
%e A363530     243: {2,2,2,2,2}
%e A363530     294: {1,2,4,4}
%e A363530     351: {2,2,2,6}
%e A363530     550: {1,3,3,5}
%e A363530     585: {2,2,3,6}
%e A363530     770: {1,3,4,5}
%e A363530     819: {2,2,4,6}
%t A363530 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A363530 Select[Range[1000],3*Total[prix[#]]==Total[Accumulate[Reverse[prix[#]]]]&]
%Y A363530 These partitions are counted by A363527.
%Y A363530 The reverse version is A363531, counted by A363526.
%Y A363530 A053632 counts compositions by weighted sum.
%Y A363530 A055396 gives minimum prime index, maximum A061395.
%Y A363530 A112798 lists prime indices, length A001222, sum A056239.
%Y A363530 A264034 counts partitions by weighted sum, reverse A358194.
%Y A363530 A304818 gives weighted sum of prime indices, row-sums of A359361.
%Y A363530 A318283 gives weighted sum of reversed prime indices, row-sums of A358136.
%Y A363530 A320387 counts multisets by weighted sum, zero-based A359678.
%Y A363530 Cf. A000041, A000720, A001221, A046660, A106529, A118914, A124010, A181819, A215366, A359362, A359755.
%K A363530 nonn
%O A363530 1,2
%A A363530 _Gus Wiseman_, Jun 12 2023