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A333328 Irregular triangle read by rows: T(n,0) = A002182(n) and T(n,k + 1) = A000005(T(n,k)), terminating at the first number which is not highly composite, n > 2.

%I A333328 #21 Apr 23 2020 22:35:09
%S A333328 4,3,6,4,3,12,6,4,3,24,8,36,9,48,10,60,12,6,4,3,120,16,180,18,240,20,
%T A333328 360,24,8,720,30,840,32,1260,36,9,1680,40,2520,48,10,5040,60,12,6,4,3,
%U A333328 7560,64,10080,72,15120,80,20160,84,25200,90,27720,96,45360,100
%N A333328 Irregular triangle read by rows: T(n,0) = A002182(n) and T(n,k + 1) = A000005(T(n,k)), terminating at the first number which is not highly composite, n > 2.
%C A333328 There are two questions related to this array: First, which rows have length greater than any previous row? Second, are there any rows which terminate at a k greater than 6?
%H A333328 James Grime and Brady Haran, <a href="https://www.youtube.com/watch?v=2JM2oImb9Qg">5040 and other Anti-Prime Numbers</a>, Numberphile video (2016).
%F A333328 T(n,0) = A002182(n), T(n,k) = A000005(T(n,k - 1)).
%e A333328 The irregular triangle T(n,k) starts:
%e A333328   n\k   0   1   2   3   4   ...
%e A333328    3:   4   3
%e A333328    4:   6   4   3
%e A333328    5:  12   6   4   3
%e A333328    6:  24   8
%e A333328    7:  36   9
%e A333328    8:  48  10
%e A333328    9:  60  12   6   4   3
%e A333328   10: 120  16
%e A333328   11: 180  18
%e A333328   12: 240  20
%e A333328   13: 360  24   8
%e A333328   ...
%o A333328 (PARI) A333328_rows(n)={my(N=Map(Mat([1,1;2,2;m=4,3])),p=2,F=[]); while(#N<n,if(numdiv(m)>p,mapput(N,m,p=numdiv(m)); my(M=List([m,q=p])); while(mapisdefined(N,q,&q),listput(M,q));print(#N", "Vec(M)); F=concat(F,Vec(M))); my(s=if(m>=720720,360360,m>=5040,2520,m>=840,420,m>=60,60,2)); until(numdiv(m+=s)>p,));F}
%Y A333328 Cf. A000005, A002182, A002183, A189394.
%K A333328 nonn,tabf
%O A333328 3,1
%A A333328 _Davis Smith_, Mar 15 2020