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

1,10

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.

Links

Table of n, a(n) for n=1..105.

Formula

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

Example

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).

Mathematica

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

Prog

(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

Crossrefs

Cf. A051731, A032742, A143112, A127093.

Sequence in context: A213731 A299197 A334196 * A328361 A301367 A334090

Adjacent sequences: A143108 A143109 A143110 * A143112 A143113 A143114

Keyword

nonn,tabl

Author

Gary W. Adamson and Mats Granvik, Jul 25 2008

Extensions

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

Status

approved