A359755 - OEIS

%I #6 Jan 16 2023 11:15:03

%S 1,2,3,4,6,7,8,10,12,15,16,18,20,24,26,28,36,40,46,48,50,52,56,62,68,

%T 74,76,86,88,92,94,106,107,118,122,124,131,134,136,142,146,152,158,

%U 164,166,173,178,188,193,194,199,202,206,214,218,226,229,236,239,254

%N Positions of first appearances in the sequence of weighted sums of prime indices (A304818).

%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 weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i.

%e The terms together with their prime indices begin:

%e     1: {}

%e     2: {1}

%e     3: {2}

%e     4: {1,1}

%e     6: {1,2}

%e     7: {4}

%e     8: {1,1,1}

%e    10: {1,3}

%e    12: {1,1,2}

%e    15: {2,3}

%e    16: {1,1,1,1}

%e    18: {1,2,2}

%e    20: {1,1,3}

%e    24: {1,1,1,2}

%t nn=1000;

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t ots[y_]:=Sum[i*y[[i]],{i,Length[y]}];

%t seq=Table[ots[primeMS[n]],{n,1,nn}];

%t Select[Range[nn],FreeQ[seq[[Range[#-1]]],seq[[#]]]&]

%Y The version for standard compositions is A089633, zero-based A359756.

%Y Positions of first appearances in A304818, reverse A318283.

%Y The zero-based version is A359675, unsorted A359676.

%Y The reverse zero-based version is A359680, unsorted A359681.

%Y This is the sorted version of A359682, reverse A359679.

%Y The reverse version is A359754.

%Y A053632 counts compositions by weighted sum.

%Y A112798 lists prime indices, length A001222, sum A056239.

%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, A124757, A243055, A358194, A359497, A359674, A359683.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jan 15 2023