cp's OEIS Frontend

A359680 Positions of first appearances in the sequence of zero-based weighted sums of reversed prime indices (A359677).

%I A359680 #5 Jan 16 2023 11:14:55
%S A359680 1,4,8,9,16,18,32,36,50,54,64,72,81,100,108,128,144,216,243,256,288,
%T A359680 300,400,432,486,512,576,600,648,729,800,864,1024,1152,1296,1350,1728,
%U A359680 1944,2048,2187,2304,2400,2916,3375,3456,3600,4096,4374,4608,4800,5184
%N A359680 Positions of first appearances in the sequence of zero-based weighted sums of reversed prime indices (A359677).
%C A359680 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 A359680 The zero-based weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} (i-1)*y_i.
%e A359680 The terms together with their prime indices begin:
%e A359680      1: {}
%e A359680      4: {1,1}
%e A359680      8: {1,1,1}
%e A359680      9: {2,2}
%e A359680     16: {1,1,1,1}
%e A359680     18: {1,2,2}
%e A359680     32: {1,1,1,1,1}
%e A359680     36: {1,1,2,2}
%e A359680     50: {1,3,3}
%e A359680     54: {1,2,2,2}
%e A359680     64: {1,1,1,1,1,1}
%e A359680     72: {1,1,1,2,2}
%e A359680     81: {2,2,2,2}
%e A359680    100: {1,1,3,3}
%e A359680    108: {1,1,2,2,2}
%e A359680    128: {1,1,1,1,1,1,1}
%t A359680 nn=1000;
%t A359680 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A359680 wts[y_]:=Sum[(i-1)*y[[i]],{i,Length[y]}];
%t A359680 seq=Table[wts[Reverse[primeMS[n]]],{n,1,nn}];
%t A359680 Select[Range[nn],FreeQ[seq[[Range[#-1]]],seq[[#]]]&]
%Y A359680 The unreversed version is A359675, unsorted A359676.
%Y A359680 Positions of first appearances in A359677, unreversed A359674.
%Y A359680 This is the sorted version of A359681.
%Y A359680 The one-based version is A359754, unsorted A359679.
%Y A359680 The unreversed one-based version is A359755, unsorted A359682.
%Y A359680 The version for standard compositions is A359756, one-based A089633.
%Y A359680 A053632 counts compositions by zero-based weighted sum.
%Y A359680 A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
%Y A359680 A124757 gives zero-based weighted sums of standard compositions, rev A231204.
%Y A359680 A304818 gives weighted sum of prime indices, reverse A318283.
%Y A359680 A320387 counts multisets by weighted sum, zero-based A359678.
%Y A359680 A358136 lists partial sums of prime indices, ranked by A358137, rev A359361.
%Y A359680 Cf. A029931, A055932, A243055, A358194, A359043, A359683.
%K A359680 nonn
%O A359680 1,2
%A A359680 _Gus Wiseman_, Jan 15 2023