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.

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, 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, 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

Offset: 1

Author

Gary W. Adamson and Mats Granvik, Jul 25 2008

Keywords

• nonn
• tabl

Comments

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

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

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.

Examples

			First few rows of the triangle:
  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, 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, 1, 2, 0, 3, 0, 0, 0, 0, 0, 6;
  ...
Example: The divisors of 12 are shown in row 12 of triangle A127093:
  (1, 2, 3, 4, 0, 6, 0, 0, 0, 0, 0, 12);
and the largest proper divisors of those terms are:
  (1, 1, 1, 2, 0, 3, 0, 0, 0, 0, 0, 6)
where the first 12 terms of A031742 (largest proper divisors of n) are:
  (1, 1, 1, 2, 1, 3, 1, 4, 3, 5, 1, 6).
		

Crossrefs

Cf. A051731, A032742, A143112, A127093.

Programs

Mathematica

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

PARI

t(n,k) = k * 0^(n % k); \\ A127093
f(n) = if(n<=1, n, n/factor(n)[1, 1]); \\ A032742
T(n,k) = f(t(n,k));
row(n) = vector(n, k, T(n,k)); \\ Michel Marcus, Dec 19 2022

PARI

T1(n,k) = 0^(n % k); \\ A051731
a2(n) = if(n==1, 1, n/factor(n)[1, 1]); \\ A032742
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

Formula

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

Extensions

Typo in data corrected and new name from existing formula by Michel Marcus, Dec 19 2022