A353745 Number of runs in the ordered prime signature of n.

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

Offset: 1

Author

Gus Wiseman, May 20 2022

Keywords

• nonn

Comments

First differs from A071625 at a(90) = 3.

First differs from A331592 at a(90) = 3.

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

Examples

			The prime indices of 630 are {1,2,2,3,4}, with multiplicities {1,2,1,1}, with runs {{1},{2},{1,1}}, so a(630) = 3.
		

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 A354233.

A001222 counts prime factors, distinct A001221.

A005361 gives product of prime signature, firsts A353500/A085629.

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

A124010 gives prime signature, sorted A118914.

A181819 gives prime shadow, with an inverse A181821.

A182850/A323014 give frequency depth, counted by A225485/A325280.

Cf. A005811, A097318, A130091, A304678, A333755, A353503, A353507, A353742.

Cf. also A329747.

Programs

Mathematica

Table[Length[Split[Last/@If[n==1,{},FactorInteger[n]]]],{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); };
runlengths(lista) = if(!#lista, lista, if(1==#lista, List([1]), my(runs=List([]), rl=1); for(i=1, #lista, if((i < #lista) && (lista[i]==lista[i+1]), rl++, listput(runs,rl); rl=1)); (runs)));
A353745(n) = #runlengths(runlengths(pis_to_runs(n))); \\ Antti Karttunen, Jan 20 2025