%I #18 Dec 20 2022 12:01:13

%S 1,1,1,1,0,1,1,1,0,2,1,0,0,0,1,1,1,1,0,0,3,1,0,0,0,0,0,1,1,1,0,2,0,0,

%T 0,4,1,0,1,0,0,0,0,0,3,1,1,0,0,1,0,0,0,0,5,1,0,0,0,0,0,0,0,0,0,1,1,1,

%U 1,2,0,3,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,1,0,0,0,0,0,0,7

%N Triangle read by rows, T(n,k) = largest proper divisor of A127093(n,k) where (largest proper divisor)(n) = A032742(n) if n>0 and 0 if n=0.

%C Previous name: A051731 * A032742 * 0^(n-k), 1 <= k <= n.

%C Row sums = A143112 = sum of (largest proper divisors of the divisors of n) = inverse Mobius transform (A051731) of A032742 (largest proper divisor of n).

%C The n-th row records the proper divisors of the divisors of n, where the divisors of n comprise triangle A127093 and the largest proper divisors of n = A032742.

%F Triangle read by rows, T(n,k) = A051731 * A032742 * 0^(n-k), 1 <= k <= n.

%e First few rows of the triangle:

%e 1;

%e 1, 1;

%e 1, 0, 1;

%e 1, 1, 0, 2;

%e 1, 0, 0, 0, 1;

%e 1, 1, 1, 0, 0, 3;

%e 1, 0, 0, 0, 0, 0, 1;

%e 1, 1, 0, 2, 0, 0, 0, 4;

%e 1, 0, 1, 0, 0, 0, 0, 0, 3;

%e 1, 1, 0, 0, 1, 0, 0, 0, 0, 5;

%e 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

%e 1, 1, 1, 2, 0, 3, 0, 0, 0, 0, 0, 6;

%e ...

%e Example: The divisors of 12 are shown in row 12 of triangle A127093:

%e (1, 2, 3, 4, 0, 6, 0, 0, 0, 0, 0, 12);

%e and the largest proper divisors of those terms are:

%e (1, 1, 1, 2, 0, 3, 0, 0, 0, 0, 0, 6)

%e where the first 12 terms of A031742 (largest proper divisors of n) are:

%e (1, 1, 1, 2, 1, 3, 1, 4, 3, 5, 1, 6).

%t Table[If[# > 1, Divisors[#][[-2]], #] &[k*Boole[Divisible[n, k]]], {n, 14}, {k, n}] (* _Michael De Vlieger_, Dec 19 2022 *)

%o (PARI) t(n,k) = k * 0^(n % k); \\ A127093

%o f(n) = if(n<=1, n, n/factor(n)[1, 1]); \\ A032742

%o T(n,k) = f(t(n,k));

%o row(n) = vector(n, k, T(n,k)); \\ _Michel Marcus_, Dec 19 2022

%o (PARI) T1(n,k) = 0^(n % k); \\ A051731

%o a2(n) = if(n==1, 1, n/factor(n)[1, 1]); \\ A032742

%o tabl(nn) = my(m1 = matrix(nn,nn,n,k,T1(n,k)), v2 = vector(nn,n,a2(n))); m1*matdiagonal(v2); \\ _Michel Marcus_, Dec 19 2022

%Y Cf. A051731, A032742, A143112, A127093.

%K nonn,tabl

%O 1,10

%A _Gary W. Adamson_ and _Mats Granvik_, Jul 25 2008

%E Typo in data corrected and new name from existing formula by _Michel Marcus_, Dec 19 2022