A353861 Number of distinct weak run-sums of the prime indices of n.

1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 3, 2, 3, 3, 5, 2, 4, 2, 4, 3, 3, 2, 4, 3, 3, 4, 4, 2, 4, 2, 6, 3, 3, 3, 4, 2, 3, 3, 4, 2, 4, 2, 4, 4, 3, 2, 5, 3, 4, 3, 4, 2, 5, 3, 5, 3, 3, 2, 4, 2, 3, 3, 7, 3, 4, 2, 4, 3, 4, 2, 5, 2, 3, 4, 4, 3, 4, 2, 5, 5, 3, 2, 4, 3, 3, 3, 5, 2, 5, 3, 4, 3, 3, 3, 6, 2, 4, 4, 5, 2, 4, 2, 5, 4, 3, 2, 5

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.

A weak run-sum of a sequence is the sum of any consecutive constant subsequence.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences related to prime indices in the factorization of n.

Example

The prime indices of 72 are {1,1,1,2,2}, with weak runs {}, {1}, {1,1}, {1,1,1}, {2}, {2,2}, which have sums 0, 1, 2, 3, 2, 4, of which 5 are distinct, so a(72) = 5.

Mathematica

Table[Length[Union@@Cases[FactorInteger[n], {p_, k_}:>Range[0, k]*PrimePi[p]]], {n, 100}]

Prog

(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); };
A353861(n) = if(1==n, n, my(pruns = pis_to_runs(n), runsum = 0, runsums = List([])); for(i=1, #pruns, listput(runsums, runsum); if((i>1) && pruns[i] == pruns[i-1], runsum += pruns[i], runsum = pruns[i])); listput(runsums, runsum); #Set(runsums)); \\ Antti Karttunen, Jan 20 2025

Crossrefs

Positions of 2's are A000040.

Positions of first appearances are A000079.

The strong version is A353835, firsts A002110.

Partitions with distinct run-sums are ranked by A353838, counted by A353837.

The strong version for compositions is A353849.

The greatest run-sum is given by A353862, least A353931.

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.

A165413 counts distinct run-lengths in binary expansion, sums A353929.

A300273 ranks collapsible partitions, counted by A275870.

A353832 represents taking run-sums of a partition, compositions A353847.

A353833 ranks partitions with all equal run-sums, counted by A304442.

A353840-A353846 pertain to partition run-sum trajectory.

A353852 ranks compositions with all distinct run-sums, counted by A353850.

Cf. A071625, A073093, A116608, A175413, A181819, A333755, A353834, A353839, A353866, A353867, A353930.

Sequence in context: A320013 A299990 A175193 * A073093 A326196 A222084

Adjacent sequences: A353858 A353859 A353860 * A353862 A353863 A353864

Keyword

nonn

Author

Gus Wiseman, May 23 2022

Extensions

Data section extended to a(108) by Antti Karttunen, Jan 20 2025

Status

approved