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%I #7 Jan 15 2023 09:51:00
%S 1,4,6,8,12,14,16,20,24,30,32,36,40,48,52,56,72,80,92,96,100,104,112,
%T 124,136,148,152,172,176,184,188,212,214,236,244,248,262,268,272,284,
%U 292,304,316,328,332,346,356,376,386,388,398,404,412,428,436,452,458
%N Positions of first appearances in the sequence of zero-based weighted sums of prime indices (A359674).
%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 6: {1,2}
%e 8: {1,1,1}
%e 12: {1,1,2}
%e 14: {1,4}
%e 16: {1,1,1,1}
%e 20: {1,1,3}
%e 24: {1,1,1,2}
%e 30: {1,2,3}
%e 32: {1,1,1,1,1}
%e 36: {1,1,2,2}
%e 40: {1,1,1,3}
%e 48: {1,1,1,1,2}
%t nn=100;
%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[primeMS[n]],{n,1,nn}];
%t Select[Range[nn],FreeQ[seq[[Range[#-1]]],seq[[#]]]&]
%Y Positions of first appearances in A359674.
%Y The unsorted version A359676.
%Y The reverse version is A359680, unsorted A359681.
%Y The reverse one-based version is A359754, unsorted A359679.
%Y The 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.
%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 A358136 lists partial sums of prime indices, ranked by A358137, rev A359361.
%Y Cf. A001248, A029931, A055932, A243055, A359043, A358194, A359360, A359497, A359677, A359683.
%K nonn
%O 1,2
%A _Gus Wiseman_, Jan 13 2023