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A111972 a(n) = Max(omega(k): 1<=k<=n), where omega(n) = A001221(n), the number of distinct prime factors of n.

%I A111972 #11 Dec 30 2021 00:24:27
%S A111972 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,
%T A111972 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,
%U A111972 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
%N A111972 a(n) = Max(omega(k): 1<=k<=n), where omega(n) = A001221(n), the number of distinct prime factors of n.
%C A111972 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.
%H A111972 Alois P. Heinz, <a href="/A111972/b111972.txt">Table of n, a(n) for n = 1..65536</a>
%H A111972 Safia Aoudjit and Djamel Berkane, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Berkane/berkane8.html">Explicit Estimates Involving the Primorial Integers and Applications</a>, J. Int. Seq., Vol. 24 (2021), Article 21.7.8.
%e A111972 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.
%p A111972 a:= proc(n) option remember; `if`(n=0, 0,
%p A111972       max(a(n-1), nops(ifactors(n)[2])))
%p A111972     end:
%p A111972 seq(a(n), n=1..105);  # _Alois P. Heinz_, Aug 19 2021
%t A111972 FoldList[Max, PrimeNu /@ Range[105]] (* _Michael De Vlieger_, Dec 29 2021 *)
%Y A111972 Cf. A001221 (omega(n)), A002110 (primorials), A000523 (Log_2(n) rounded down), A001222 (Omega(n), also known as bigomega(n)).
%K A111972 nonn
%O A111972 1,6
%A A111972 _Rick L. Shepherd_, Aug 24 2005