OFFSET
1,2
COMMENTS
A number's (unordered) prime signature (row n of A118914) is the multiset of positive exponents in its prime factorization.
EXAMPLE
The terms together with their prime indices begin:
1: {}
2: {1}
6: {1,2}
30: {1,2,3}
49: {4,4}
152: {1,1,1,8}
210: {1,2,3,4}
513: {2,2,2,8}
1444: {1,1,8,8}
1776: {1,1,1,1,2,12}
1952: {1,1,1,1,1,18}
2310: {1,2,3,4,5}
2375: {3,3,3,8}
2664: {1,1,1,2,2,12}
2760: {1,1,1,2,3,9}
2960: {1,1,1,1,3,12}
For example, the prime indices of 2760 are {1,1,1,2,3,9}. The signature is (3,1,1,1), with mean 3/2. The first differences of 0-prepended prime indices are (1,0,0,1,1,6), with mean also 3/2. So 2760 is in the sequence.
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[1000], Mean[Length/@Split[prix[#]]] == Mean[Differences[Prepend[prix[#], 0]]]&]
CROSSREFS
For median instead of mean we have A360681.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 19 2023
STATUS
approved