Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.
%I #5 Jan 16 2023 11:14:55
%S 1,4,8,9,16,18,32,36,50,54,64,72,81,100,108,128,144,216,243,256,288,
%T 300,400,432,486,512,576,600,648,729,800,864,1024,1152,1296,1350,1728,
%U 1944,2048,2187,2304,2400,2916,3375,3456,3600,4096,4374,4608,4800,5184
%N Positions of first appearances in the sequence of zero-based weighted sums of reversed prime indices (A359677).
%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 8: {1,1,1}
%e 9: {2,2}
%e 16: {1,1,1,1}
%e 18: {1,2,2}
%e 32: {1,1,1,1,1}
%e 36: {1,1,2,2}
%e 50: {1,3,3}
%e 54: {1,2,2,2}
%e 64: {1,1,1,1,1,1}
%e 72: {1,1,1,2,2}
%e 81: {2,2,2,2}
%e 100: {1,1,3,3}
%e 108: {1,1,2,2,2}
%e 128: {1,1,1,1,1,1,1}
%t nn=1000;
%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,nn}];
%t Select[Range[nn],FreeQ[seq[[Range[#-1]]],seq[[#]]]&]
%Y The unreversed version is A359675, unsorted A359676.
%Y Positions of first appearances in A359677, unreversed A359674.
%Y This is the sorted version of A359681.
%Y The one-based version is A359754, unsorted A359679.
%Y The unreversed one-based version is A359755, unsorted A359682.
%Y The version for standard compositions is A359756, one-based A089633.
%Y A053632 counts compositions by zero-based weighted sum.
%Y A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
%Y A124757 gives zero-based weighted sums 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 A358136 lists partial sums of prime indices, ranked by A358137, rev A359361.
%Y Cf. A029931, A055932, A243055, A358194, A359043, A359683.
%K nonn
%O 1,2
%A _Gus Wiseman_, Jan 15 2023