A353745 - OEIS

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

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OFFSET

1,12

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.

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.

EXAMPLE

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.

MATHEMATICA

Table[Length[Split[Last/@If[n==1, {}, FactorInteger[n]]]], {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); };

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

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.

Sequence in context: A003652 A071625 A331592 * A309004 A355382 A304779