cp's OEIS Frontend

A359675 Positions of first appearances in the sequence of zero-based weighted sums of prime indices (A359674).

%I A359675 #7 Jan 15 2023 09:51:00
%S A359675 1,4,6,8,12,14,16,20,24,30,32,36,40,48,52,56,72,80,92,96,100,104,112,
%T A359675 124,136,148,152,172,176,184,188,212,214,236,244,248,262,268,272,284,
%U A359675 292,304,316,328,332,346,356,376,386,388,398,404,412,428,436,452,458
%N A359675 Positions of first appearances in the sequence of zero-based weighted sums of prime indices (A359674).
%C A359675 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 A359675 The zero-based weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} (i-1)*y_i.
%e A359675 The terms together with their prime indices begin:
%e A359675    1: {}
%e A359675    4: {1,1}
%e A359675    6: {1,2}
%e A359675    8: {1,1,1}
%e A359675   12: {1,1,2}
%e A359675   14: {1,4}
%e A359675   16: {1,1,1,1}
%e A359675   20: {1,1,3}
%e A359675   24: {1,1,1,2}
%e A359675   30: {1,2,3}
%e A359675   32: {1,1,1,1,1}
%e A359675   36: {1,1,2,2}
%e A359675   40: {1,1,1,3}
%e A359675   48: {1,1,1,1,2}
%t A359675 nn=100;
%t A359675 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A359675 wts[y_]:=Sum[(i-1)*y[[i]],{i,Length[y]}];
%t A359675 seq=Table[wts[primeMS[n]],{n,1,nn}];
%t A359675 Select[Range[nn],FreeQ[seq[[Range[#-1]]],seq[[#]]]&]
%Y A359675 Positions of first appearances in A359674.
%Y A359675 The unsorted version A359676.
%Y A359675 The reverse version is A359680, unsorted A359681.
%Y A359675 The reverse one-based version is A359754, unsorted A359679.
%Y A359675 The one-based version is A359755, unsorted A359682.
%Y A359675 The version for standard compositions is A359756, one-based A089633.
%Y A359675 A053632 counts compositions by zero-based weighted sum.
%Y A359675 A112798 lists prime indices, length A001222, sum A056239.
%Y A359675 A124757 gives zero-based weighted sum of standard compositions, rev A231204.
%Y A359675 A304818 gives weighted sum of prime indices, reverse A318283.
%Y A359675 A320387 counts multisets by weighted sum, zero-based A359678.
%Y A359675 A358136 lists partial sums of prime indices, ranked by A358137, rev A359361.
%Y A359675 Cf. A001248, A029931, A055932, A243055, A359043, A358194, A359360, A359497, A359677, A359683.
%K A359675 nonn
%O A359675 1,2
%A A359675 _Gus Wiseman_, Jan 13 2023