cp's OEIS Frontend

A363531 Heinz numbers of integer partitions such that 3*(sum) = (reverse-weighted sum).

%I A363531 #5 Jun 12 2023 08:42:37
%S A363531 1,32,144,216,243,672,1008,1350,2176,2250,2520,2673,3125,3969,4160,
%T A363531 4200,5940,6240,6615,7344,7424,7744,8262,9261,9800,9900,10400,11616,
%U A363531 12250,12312,12375,13104,13720,14720,14742,16767,16807,17150,19360,21840,22080,23100
%N A363531 Heinz numbers of integer partitions such that 3*(sum) = (reverse-weighted sum).
%C A363531 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 A363531 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 A363531 A056239(a(n)) = A318283(a(n))/3.
%e A363531 The terms together with their prime indices begin:
%e A363531       1: {}
%e A363531      32: {1,1,1,1,1}
%e A363531     144: {1,1,1,1,2,2}
%e A363531     216: {1,1,1,2,2,2}
%e A363531     243: {2,2,2,2,2}
%e A363531     672: {1,1,1,1,1,2,4}
%e A363531    1008: {1,1,1,1,2,2,4}
%e A363531    1350: {1,2,2,2,3,3}
%e A363531    2176: {1,1,1,1,1,1,1,7}
%e A363531    2250: {1,2,2,3,3,3}
%e A363531    2520: {1,1,1,2,2,3,4}
%e A363531    2673: {2,2,2,2,2,5}
%e A363531    3125: {3,3,3,3,3}
%e A363531    3969: {2,2,2,2,4,4}
%e A363531    4160: {1,1,1,1,1,1,3,6}
%t A363531 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A363531 Select[Range[1000],3*Total[prix[#]]==Total[Accumulate[prix[#]]]&]
%Y A363531 These partitions are counted by A363526.
%Y A363531 The non-reverse version is A363530, counted by A363527.
%Y A363531 A053632 counts compositions by weighted sum.
%Y A363531 A055396 gives minimum prime index, maximum A061395.
%Y A363531 A112798 lists prime indices, length A001222, sum A056239.
%Y A363531 A264034 counts partitions by weighted sum, reverse A358194.
%Y A363531 A304818 gives weighted sum of prime indices, row-sums of A359361.
%Y A363531 A318283 gives weighted sum of reversed prime indices, row-sums of A358136.
%Y A363531 A320387 counts multisets by weighted sum, zero-based A359678.
%Y A363531 Cf. A000041, A000720, A001221, A046660, A106529, A118914, A124010, A181819, A215366, A359362, A359755.
%K A363531 nonn
%O A363531 1,2
%A A363531 _Gus Wiseman_, Jun 12 2023