A208460 Triangle read by rows: T(n,k) = n minus the k-th proper divisor of n.

1, 2, 3, 2, 4, 5, 4, 3, 6, 7, 6, 4, 8, 6, 9, 8, 5, 10, 11, 10, 9, 8, 6, 12, 13, 12, 7, 14, 12, 10, 15, 14, 12, 8, 16, 17, 16, 15, 12, 9, 18, 19, 18, 16, 15, 10, 20, 18, 14, 21, 20, 11, 22, 23, 22, 21, 20, 18, 16, 12, 24, 20, 25, 24, 13, 26, 24, 18, 27, 26, 24

Offset: 2

Author

Omar E. Pol, Feb 28 2012

Keywords

• nonn
• tabf

Comments

Conjecture: one of the divisors of T(n,k) is also the k-th divisor of n. In a diagram of the structure of divisors of the natural numbers (see link) the mentioned divisors of the elements of row n are located on a straight line to 45 degrees from the vertical straight line that contains the divisors of n, therefore the divisors of n are predictable.

Examples

			Written as a triangle starting from n = 2:
1;
2;
3, 2;
4;
5, 4, 3;
6;
7, 6, 4;
8, 6;
9, 8, 5;
10;
11, 10, 9, 8, 6;
12;
		

Links

• Alois P. Heinz, Rows n = 2..1540, flattened
• Omar E. Pol, Illustration of the structure of divisors of the natural numbers, for n = 1..16

Crossrefs

Column 1 is A000027. Row n has length A032741(n). Row sums give the positives A094471. Right border is A060681.

Cf. A000005, A027750, A027751.

Programs

Maple

with (numtheory):
T:= n-> map(x-> n-x, sort([(divisors(n) minus {n})[]]))[]:
seq (T(n), n=2..50); # Alois P. Heinz, Apr 11 2012

Mathematica

T[n_] := Most[n-Divisors[n]]; Table[T[n], {n, 2, 50}] // Flatten (* Jean-François Alcover, Feb 21 2017 *)

Formula

T(n,k) = n - A027751(n,k).