A216621 Triangle read by rows, n >= 1, 1 <= k <= n, T(n,k) = Sum_{c|n,d|k} phi(gcd(c,d)).

1, 2, 4, 2, 4, 5, 3, 6, 6, 10, 2, 4, 4, 6, 7, 4, 8, 10, 12, 8, 20, 2, 4, 4, 6, 4, 8, 9, 4, 8, 8, 14, 8, 16, 8, 22, 3, 6, 8, 9, 6, 16, 6, 12, 17, 4, 8, 8, 12, 14, 16, 8, 16, 12, 28, 2, 4, 4, 6, 4, 8, 4, 8, 6, 8, 13, 6, 12, 15, 20, 12, 30, 12, 28, 24, 24, 12

Offset: 1

Author

Peter Luschny, Sep 12 2012

Keywords

• nonn
• tabl

Comments

This is the lower triangular array of A216620, which is the main entry for this sequence.

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

T(n,n) = A060648(n) = sum{d|n} Dedekind_Psi(d).

Examples

			The first rows of the triangle are:
  1;
  2,  4;
  2,  4,  5;
  3,  6,  6, 10;
  2,  4,  4,  6,  7;
  4,  8, 10, 12,  8, 20;
  2,  4,  4,  6,  4,  8,  9;
  4,  8,  8, 14,  8, 16,  8, 22;
  3,  6,  8,  9,  6, 16,  6, 12, 17;
  4,  8,  8, 12, 14, 16,  8, 16, 12, 28;
  2,  4,  4,  6,  4,  8,  4,  8,  6,  8, 13;
		

Links

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

Crossrefs

Cf. A216620, A216622, A216623, A216624, A216625, A216626, A216627.

Programs

Maple

with(numtheory):
T:= (n, k)-> add(add(phi(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[ EulerPhi[GCD[c, d]], {c, Divisors[n]}, {d, Divisors[k]}]; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)

Sage

for n in (1..9): [A216620(n,k) for k in (1..n)]