A216626 Square array read by antidiagonals, T(n,k) = sum_{c|n,d|k} lcm(c,d) for n>=1, k>=1.

1, 3, 3, 4, 7, 4, 7, 12, 12, 7, 6, 15, 10, 15, 6, 12, 18, 28, 28, 18, 12, 8, 28, 24, 27, 24, 28, 8, 15, 24, 30, 42, 42, 30, 24, 15, 13, 31, 32, 60, 16, 60, 32, 31, 13, 18, 39, 60, 56, 72, 72, 56, 60, 39, 18, 12, 42, 28, 51, 48, 70, 48, 51, 28, 42, 12, 28, 36

Offset: 1

Author

Peter Luschny, Sep 12 2012

Keywords

• nonn
• tabl

Comments

T(n,n) = A064950(n) = sum_{d|n} d*tau(d^2).

T(n,1) = T(1,n) = A000203(n) = sigma(n).

T(n,2) = T(2,n) = A062731(n) = sigma(2*n).

T(n+1,n) = A083539(n) = sigma(n+1)*sigma(n).

T(prime(n),1) = A008864(n) = prime(n)+1.

Examples

			[-----1---2---3----4----5----6----7----8----9---10---11---12]
[ 1]  1,  3,  4,   7,   6,  12,   8,  15,  13,  18,  12,  28
[ 2]  3,  7, 12,  15,  18,  28,  24,  31,  39,  42,  36,  60
[ 3]  4, 12, 10,  28,  24,  30,  32,  60,  28,  72,  48,  70
[ 4]  7, 15, 28,  27,  42,  60,  56,  51,  91,  90,  84, 108
[ 5]  6, 18, 24,  42,  16,  72,  48,  90,  78,  48,  72, 168
[ 6] 12, 28, 30,  60,  72,  70,  96, 124,  84, 168, 144, 150
[ 7]  8, 24, 32,  56,  48,  96,  22, 120, 104, 144,  96, 224
[ 8] 15, 31, 60,  51,  90, 124, 120,  83, 195, 186, 180, 204
[ 9] 13, 39, 28,  91,  78,  84, 104, 195,  55, 234, 156, 196
[10] 18, 42, 72,  90,  48, 168, 144, 186, 234, 112, 216, 360
[11] 12, 36, 48,  84,  72, 144,  96, 180, 156, 216,  34, 336
[12] 28, 60, 70, 108, 168, 150, 224, 204, 196, 360, 336, 270
.
Displayed as a triangular array:
    1;
    3,  3;
    4,  7,  4;
    7, 12, 12,  7;
    6, 15, 10, 15,  6;
   12, 18, 28, 28, 18, 12;
    8, 28, 24, 27, 24, 28,  8;
   15, 24, 30, 42, 42, 30, 24, 15;
   13, 31, 32, 60, 16, 60, 32, 31, 13;
		

Links

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

Crossrefs

Cf. A216620, A216621, A216622, A216623, A216624, A216625, A216627.

Programs

Maple

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

Mathematica

T[n_, k_] := Sum[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, Mar 25 2014 *)

Sage

def A216626(n, k) :
    cp = cartesian_product([divisors(n), divisors(k)])
    return reduce(lambda x,y: x+y, map(lcm, cp))
for n in (1..12): [A216626(n,k) for k in (1..12)]