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Heinz numbers of integer partitions such that 3*(sum) = (reverse-weighted sum).
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%I #5 Jun 12 2023 08:42:37

%S 1,32,144,216,243,672,1008,1350,2176,2250,2520,2673,3125,3969,4160,

%T 4200,5940,6240,6615,7344,7424,7744,8262,9261,9800,9900,10400,11616,

%U 12250,12312,12375,13104,13720,14720,14742,16767,16807,17150,19360,21840,22080,23100

%N Heinz numbers of integer partitions such that 3*(sum) = (reverse-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. The reverse-weighted sum is the weighted sum of the reverse, also the sum of partial sums. For example, the weighted sum of (4,2,2,1) is 1*4 + 2*2 + 3*2 + 4*1 = 18 and the reverse-weighted sum is 4*4 + 3*2 + 2*2 + 1*1 = 27.

%F A056239(a(n)) = A318283(a(n))/3.

%e The terms together with their prime indices begin:

%e 1: {}

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

%e 144: {1,1,1,1,2,2}

%e 216: {1,1,1,2,2,2}

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

%e 672: {1,1,1,1,1,2,4}

%e 1008: {1,1,1,1,2,2,4}

%e 1350: {1,2,2,2,3,3}

%e 2176: {1,1,1,1,1,1,1,7}

%e 2250: {1,2,2,3,3,3}

%e 2520: {1,1,1,2,2,3,4}

%e 2673: {2,2,2,2,2,5}

%e 3125: {3,3,3,3,3}

%e 3969: {2,2,2,2,4,4}

%e 4160: {1,1,1,1,1,1,3,6}

%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[prix[#]]]&]

%Y These partitions are counted by A363526.

%Y The non-reverse version is A363530, counted by A363527.

%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