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%I #7 Jan 15 2023 09:51:40
%S 1,4,9,8,18,50,16,36,100,54,32,72,81,108,300,64,144,400,216,600,243,
%T 128,288,800,432,486,1350,648,256,576,729,864,2400,3375,1296,3600,512,
%U 1152,1944,1728,4800,9000,2187,2916,8100,1024,2304,6400,3456,4374,12150
%N Least positive integer whose reversed (weakly decreasing) prime indices have zero-based weighted sum (A359677) equal to n.
%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C The zero-based weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} (i-1)*y_i.
%e The terms together with their prime indices begin:
%e 1: {}
%e 4: {1,1}
%e 9: {2,2}
%e 8: {1,1,1}
%e 18: {1,2,2}
%e 50: {1,3,3}
%e 16: {1,1,1,1}
%e 36: {1,1,2,2}
%e 100: {1,1,3,3}
%e 54: {1,2,2,2}
%e 32: {1,1,1,1,1}
%e 72: {1,1,1,2,2}
%e 81: {2,2,2,2}
%e 108: {1,1,2,2,2}
%e 300: {1,1,2,3,3}
%t nn=20;
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t wts[y_]:=Sum[(i-1)*y[[i]],{i,Length[y]}];
%t seq=Table[wts[Reverse[primeMS[n]]],{n,1,Prime[nn]^2}];
%t Table[Position[seq,k][[1,1]],{k,0,nn}]
%Y The unreversed version is A359676.
%Y First position of n in A359677, reverse A359674.
%Y The one-based version is A359679, sorted A359754.
%Y The sorted version is A359680, reverse A359675.
%Y The unreversed one-based version is A359682, sorted A359755.
%Y A053632 counts compositions by zero-based weighted sum.
%Y A112798 lists prime indices, length A001222, sum A056239.
%Y A124757 gives zero-based weighted sum of standard compositions, rev A231204.
%Y A304818 gives weighted sum of prime indices, reverse A318283.
%Y A320387 counts multisets by weighted sum, zero-based A359678.
%Y Cf. A001248, A029931, A055932, A089633, A243055, A359043, A358194, A359360, A359361, A359497, A359683.
%K nonn
%O 0,2
%A _Gus Wiseman_, Jan 15 2023
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