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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.
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.
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
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