A216622 Square array read by antidiagonals: T(n,k) = Sum_{c|n, d|k} phi(lcm(c,d)) for n >= 1, k >= 1.

1, 2, 2, 3, 4, 3, 4, 6, 6, 4, 5, 8, 7, 8, 5, 6, 10, 12, 12, 10, 6, 7, 12, 15, 14, 15, 12, 7, 8, 14, 14, 20, 20, 14, 14, 8, 9, 16, 21, 24, 13, 24, 21, 16, 9, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 11, 20, 19, 26, 35, 28, 35, 26, 19, 20, 11, 12, 22, 30, 36, 40

Offset: 1

Author

Peter Luschny, Sep 12 2012

Keywords

• nonn
• tabl

Comments

T(n,n) = A062380(n) = Sum_{d|n} phi(d)*tau(d^2).

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

T(n,2) = T(2,n) = A005843(n) = 2*n.

T(n+1,n) = A002378(n) = (n+1)*n.

T(prime(n),1) = A000040(n) = prime(n).

T(prime(n),prime(n)) = 3*prime(n)-2.

Examples

			[-----1---2---3---4---5---6---7---8---9---10---11---12]
[ 1]  1,  2,  3,  4,  5,  6,  7,  8,  9,  10,  11,  12
[ 2]  2,  4,  6,  8, 10, 12, 14, 16, 18,  20,  22,  24
[ 3]  3,  6,  7, 12, 15, 14, 21, 24, 19,  30,  33,  28
[ 4]  4,  8, 12, 14, 20, 24, 28, 26, 36,  40,  44,  42
[ 5]  5, 10, 15, 20, 13, 30, 35, 40, 45,  26,  55,  60
[ 6]  6, 12, 14, 24, 30, 28, 42, 48, 38,  60,  66,  56
[ 7]  7, 14, 21, 28, 35, 42, 19, 56, 63,  70,  77,  84
[ 8]  8, 16, 24, 26, 40, 48, 56, 42, 72,  80,  88,  78
[ 9]  9, 18, 19, 36, 45, 38, 63, 72, 37,  90,  99,  76
[10] 10, 20, 30, 40, 26, 60, 70, 80, 90,  52, 110, 120
[11] 11, 22, 33, 44, 55, 66, 77, 88, 99, 110,  31, 132
[12] 12, 24, 28, 42, 60, 56, 84, 78, 76, 120, 132,  98
.
Displayed as a triangular array:
   1,
   2,  2,
   3,  4,  3,
   4,  6,  6,  4,
   5,  8,  7,  8,  5,
   6, 10, 12, 12, 10,  6,
   7, 12, 15, 14, 15, 12,  7,
   8, 14, 14, 20, 20, 14, 14,  8,
   9, 16, 21, 24, 13, 24, 21, 16,  9,
		

Links

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

Crossrefs

Cf. A216620, A216621, A216623, 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, 1+d-n), n=1..d), d=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+1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)

Sage

def A216622(n, k) :
    cp = cartesian_product([divisors(n), divisors(k)])
    return reduce(lambda x,y: x+y, map(euler_phi, map(lcm, cp)))
for n in (1..12): [A216622(n,k) for k in (1..12)]