A216623 Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = Sum_{c|n,d|k} phi(lcm(c,d)).

1, 2, 4, 3, 6, 7, 4, 8, 12, 14, 5, 10, 15, 20, 13, 6, 12, 14, 24, 30, 28, 7, 14, 21, 28, 35, 42, 19, 8, 16, 24, 26, 40, 48, 56, 42, 9, 18, 19, 36, 45, 38, 63, 72, 37, 10, 20, 30, 40, 26, 60, 70, 80, 90, 52, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 31, 12, 24

Offset: 1

Author

Peter Luschny, Sep 12 2012

Keywords

• nonn
• tabl

Comments

This is the lower triangular array of A216622, which is the main entry for this sequence.

T(n,1) = A000027(n).

T(n,n) = A062380(n).

Examples

			The first rows of the triangle are:
1,
2,  4,
3,  6,  7,
4,  8, 12, 14,
5, 10, 15, 20, 13,
6, 12, 14, 24, 30, 28,
7, 14, 21, 28, 35, 42, 19,
8, 16, 24, 26, 40, 48, 56, 42,
9, 18, 19, 36, 45, 38, 63, 72, 37,
		

Links

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

Crossrefs

Cf. A216620, A216621, A216622, A216624, A216625, A216626, A216627.

Programs

Maple

with(numtheory):
T:= (n, k)-> add(add(phi(ilcm(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[ LCM[c, d]], {c, Divisors[n]}, {d, Divisors[k]}]; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 23 2013 *)

Sage

# uses[A216622]
for n in (1..9): [A216622(n,k) for k in (1..n)]