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A143111 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.

%I A143111 #18 Dec 20 2022 12:01:13
%S A143111 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 A143111 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 A143111 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 A143111 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 A143111 Previous name: A051731 * A032742 * 0^(n-k), 1 <= k <= n.
%C A143111 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 A143111 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 A143111 Triangle read by rows, T(n,k) = A051731 * A032742 * 0^(n-k), 1 <= k <= n.
%e A143111 First few rows of the triangle:
%e A143111   1;
%e A143111   1, 1;
%e A143111   1, 0, 1;
%e A143111   1, 1, 0, 2;
%e A143111   1, 0, 0, 0, 1;
%e A143111   1, 1, 1, 0, 0, 3;
%e A143111   1, 0, 0, 0, 0, 0, 1;
%e A143111   1, 1, 0, 2, 0, 0, 0, 4;
%e A143111   1, 0, 1, 0, 0, 0, 0, 0, 3;
%e A143111   1, 1, 0, 0, 1, 0, 0, 0, 0, 5;
%e A143111   1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e A143111   1, 1, 1, 2, 0, 3, 0, 0, 0, 0, 0, 6;
%e A143111   ...
%e A143111 Example: The divisors of 12 are shown in row 12 of triangle A127093:
%e A143111   (1, 2, 3, 4, 0, 6, 0, 0, 0, 0, 0, 12);
%e A143111 and the largest proper divisors of those terms are:
%e A143111   (1, 1, 1, 2, 0, 3, 0, 0, 0, 0, 0, 6)
%e A143111 where the first 12 terms of A031742 (largest proper divisors of n) are:
%e A143111   (1, 1, 1, 2, 1, 3, 1, 4, 3, 5, 1, 6).
%t A143111 Table[If[# > 1, Divisors[#][[-2]], #] &[k*Boole[Divisible[n, k]]], {n, 14}, {k, n}] (* _Michael De Vlieger_, Dec 19 2022 *)
%o A143111 (PARI) t(n,k) = k * 0^(n % k); \\ A127093
%o A143111 f(n) = if(n<=1, n, n/factor(n)[1, 1]); \\ A032742
%o A143111 T(n,k) = f(t(n,k));
%o A143111 row(n) = vector(n, k, T(n,k)); \\ _Michel Marcus_, Dec 19 2022
%o A143111 (PARI) T1(n,k) = 0^(n % k); \\ A051731
%o A143111 a2(n) = if(n==1, 1, n/factor(n)[1, 1]); \\ A032742
%o A143111 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 A143111 Cf. A051731, A032742, A143112, A127093.
%K A143111 nonn,tabl
%O A143111 1,10
%A A143111 _Gary W. Adamson_ and _Mats Granvik_, Jul 25 2008
%E A143111 Typo in data corrected and new name from existing formula by _Michel Marcus_, Dec 19 2022