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A359755
Positions of first appearances in the sequence of weighted sums of prime indices (A304818).
22
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, 74, 76, 86, 88, 92, 94, 106, 107, 118, 122, 124, 131, 134, 136, 142, 146, 152, 158, 164, 166, 173, 178, 188, 193, 194, 199, 202, 206, 214, 218, 226, 229, 236, 239, 254
OFFSET
1,2
COMMENTS
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.
The weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i.
EXAMPLE
The terms together with their prime indices begin:
1: {}
2: {1}
3: {2}
4: {1,1}
6: {1,2}
7: {4}
8: {1,1,1}
10: {1,3}
12: {1,1,2}
15: {2,3}
16: {1,1,1,1}
18: {1,2,2}
20: {1,1,3}
24: {1,1,1,2}
MATHEMATICA
nn=1000;
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
ots[y_]:=Sum[i*y[[i]], {i, Length[y]}];
seq=Table[ots[primeMS[n]], {n, 1, nn}];
Select[Range[nn], FreeQ[seq[[Range[#-1]]], seq[[#]]]&]
CROSSREFS
The version for standard compositions is A089633, zero-based A359756.
Positions of first appearances in A304818, reverse A318283.
The zero-based version is A359675, unsorted A359676.
The reverse zero-based version is A359680, unsorted A359681.
This is the sorted version of A359682, reverse A359679.
The reverse version is A359754.
A053632 counts compositions by weighted sum.
A112798 lists prime indices, length A001222, sum A056239.
A320387 counts multisets by weighted sum, zero-based A359678.
A358136 lists partial sums of prime indices, ranked by A358137, rev A359361.
Sequence in context: A336488 A221178 A080389 * A226038 A181062 A338031
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 15 2023
STATUS
approved