A111972 - OEIS

A111972

a(n) = Max(omega(k): 1<=k<=n), where omega(n) = A001221(n), the number of distinct prime factors of n.

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

COMMENTS

This sequence has the same relationship to A001221 as A000523 has to A001222. Also, for n>=1, n-1 occurs as A002110(n)-A002110(n-1) consecutive terms beginning with term a(A002110(n-1)), where A002110 is the primorials; i.e. the frequencies of occurrence are the first differences (1,4,24,180,...) of the primorials.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..65536

Safia Aoudjit and Djamel Berkane, Explicit Estimates Involving the Primorial Integers and Applications, J. Int. Seq., Vol. 24 (2021), Article 21.7.8.

EXAMPLE

a(7)=2 because omega(1)=0, omega(2)=omega(3)=omega(4)=omega(5)=omega(7)=1 and omega(6)=2 (as 6=2*3), so 2 is the maximum.

MAPLE

a:= proc(n) option remember; `if`(n=0, 0,

max(a(n-1), nops(ifactors(n)[2])))

end:

seq(a(n), n=1..105); # Alois P. Heinz, Aug 19 2021

MATHEMATICA

FoldList[Max, PrimeNu /@ Range[105]] (* Michael De Vlieger, Dec 29 2021 *)