A353835 Number of distinct run-sums of the prime indices of n.

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 1, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 3

Offset: 1

Author

Gus Wiseman, May 23 2022

Keywords

• nonn

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).

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)
• Index entries for sequences related to prime indices in the factorization of n.

Crossrefs

Positions of first appearances are A002110.

A version for binary expansion is A165413.

Positions of 0's and 1's are A353833, nonprime A353834, counted by A304442.

The case of all distinct run-sums is ranked by A353838, counted by A353837.

The version for compositions is A353849.

The weak version is A353861.

A001222 counts prime factors, distinct A001221.

A005811 counts runs in binary expansion.

A056239 adds up prime indices, row sums of A112798 and A296150.

A124010 gives prime signature, sorted A118914.

A300273 ranks collapsible partitions, counted by A275870.

A351014 counts distinct runs in standard compositions.

A353832 represents the operation of taking run-sums of a partition.

A353840-A353846 pertain to partition run-sum trajectory.

A353862 gives greatest run-sum of prime indices, least A353931.

A353866 ranks rucksack partitions, counted by A353864.

Cf. A071625, A073093, A116608, A175413, A181819, A333755, A353839, A353848, A353850, A353852, A353867.

Programs

Mathematica

Table[Length[Union[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>PrimePi[p]*k]]],{n,100}]

PARI

pis_to_runs(n) = { my(runs=List([]), f=factor(n)); for(i=1,#f~,while(f[i,2], listput(runs,primepi(f[i,1])); f[i,2]--)); (runs); };
A353832(n) = if(1==n,n,my(pruns = pis_to_runs(n), m=1, runsum=pruns[1]); for(i=2,#pruns,if(pruns[i] == pruns[i-1], runsum += pruns[i], m *= prime(runsum); runsum = pruns[i])); (m*prime(runsum)));
A353835(n) = omega(A353832(n)); \\ Antti Karttunen, Jan 20 2025

Formula

a(n) = A001221(A353832(n)). [From formula section of A353832] - Antti Karttunen, Jan 20 2025

Extensions

Data section extended to a(105) by Antti Karttunen, Jan 20 2025