A216625 Triangle read by rows, n >= 1, 1 <= k <= n, T(n,k) = Sum_{c|n,d|k} gcd(c,d).

1, 2, 5, 2, 4, 6, 3, 8, 6, 15, 2, 4, 4, 6, 8, 4, 10, 12, 16, 8, 30, 2, 4, 4, 6, 4, 8, 10, 4, 11, 8, 22, 8, 22, 8, 37, 3, 6, 10, 9, 6, 20, 6, 12, 23, 4, 10, 8, 16, 16, 20, 8, 22, 12, 40, 2, 4, 4, 6, 4, 8, 4, 8, 6, 8, 14, 6, 16, 18, 30, 12, 48, 12, 44, 30, 32

Offset: 1

Author

Peter Luschny, Sep 12 2012

Keywords

• nonn
• tabl

Comments

This is the lower triangular array of A216624, which is the main entry for this sequence.

T(n,1) = A000005(n) = tau(n).

T(n,n) = A060724(n) = Sum_{d|n} d*tau((n/d)^2).

Examples

			The first rows of the triangle are:
  1;
  2,  5;
  2,  4,  6;
  3,  8,  6, 15;
  2,  4,  4,  6,  8;
  4, 10, 12, 16,  8, 30;
  2,  4,  4,  6,  4,  8, 10;
  4, 11,  8, 22,  8, 22,  8, 37;
  3,  6, 10,  9,  6, 20,  6, 12, 23;
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened

Crossrefs

Cf. A216620, A216621, A216622, A216623, A216624, A216626, A216627.

Programs

Maple

with(numtheory):
T:= (n, k)-> add(add(igcd(c, d), c=divisors(n)), d=divisors(k)):
seq (seq (T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Sep 12 2012

Mathematica

T[n_, k_] := Sum[GCD[c, d], {c, Divisors[n]}, {d, Divisors[k]}]; Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 25 2014 *)

Sage

for n in (1..9): [A216624(n,k) for k in (1..n)]