OFFSET
1,6
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 sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. For example, the runs of (2,2,1,1,1,3,2,2) are (2,2), (1,1,1), (3), (2,2), with sums (4,3,3,4).
LINKS
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)
EXAMPLE
The prime indices of 3780 are {1,1,2,2,2,3,4}, with distinct run-sums {2,3,4,6}, so a(3780) = 4.
The prime indices of 8820 are {1,1,2,2,3,4,4}, with distinct run-sums {2,3,4,8}, so a(8820) = 4.
The prime indices of 13860 are {1,1,2,2,3,4,5}, with distinct run-sums {2,3,4,5}, so a(13860) = 4.
The prime indices of 92400 are {1,1,1,1,2,3,3,4,5}, with distinct run-sums {2,4,5,6}, so a(92400) = 4.
MATHEMATICA
Table[Length[Union[Cases[If[n==1, {}, FactorInteger[n]], {p_, k_}:>PrimePi[p]*k]]], {n, 100}]
CROSSREFS
Positions of first appearances are A002110.
A version for binary expansion is A165413.
The version for compositions is A353849.
The weak version is A353861.
A005811 counts runs in binary expansion.
A351014 counts distinct runs in standard compositions.
A353832 represents the operation of taking run-sums of a partition.
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 23 2022
STATUS
approved