A216620 - OEIS

A216620

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

1, 2, 2, 2, 4, 2, 3, 4, 4, 3, 2, 6, 5, 6, 2, 4, 4, 6, 6, 4, 4, 2, 8, 4, 10, 4, 8, 2, 4, 4, 10, 6, 6, 10, 4, 4, 3, 8, 4, 12, 7, 12, 4, 8, 3, 4, 6, 8, 6, 8, 8, 6, 8, 6, 4, 2, 8, 8, 14, 4, 20, 4, 14, 8, 8, 2, 6, 4, 8, 9, 8, 8, 8, 8, 9, 8, 4, 6, 2, 12, 4, 12, 6

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

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

T(n,2) = T(2,n) = A062011(n) = 2*tau(n).

T(n+1,n) = A092517(n) = tau(n+1)*tau(n).

T(prime(n),1) = A007395(n) = 2.

T(prime(n),prime(n)) = A052147(n) = prime(n)+2.

LINKS

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

EXAMPLE

[----1---2---3---4---5---6---7---8---9--10--11--12]

[ 1] 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6

[ 2] 2, 4, 4, 6, 4, 8, 4, 8, 6, 8, 4, 12

[ 3] 2, 4, 5, 6, 4, 10, 4, 8, 8, 8, 4, 15

[ 4] 3, 6, 6, 10, 6, 12, 6, 14, 9, 12, 6, 20

[ 5] 2, 4, 4, 6, 7, 8, 4, 8, 6, 14, 4, 12

[ 6] 4, 8, 10, 12, 8, 20, 8, 16, 16, 16, 8, 30

[ 7] 2, 4, 4, 6, 4, 8, 9, 8, 6, 8, 4, 12

[ 8] 4, 8, 8, 14, 8, 16, 8, 22, 12, 16, 8, 28

[ 9] 3, 6, 8, 9, 6, 16, 6, 12, 17, 12, 6, 24

[10] 4, 8, 8, 12, 14, 16, 8, 16, 12, 28, 8, 24

[11] 2, 4, 4, 6, 4, 8, 4, 8, 6, 8, 13, 12

[12] 6, 12, 15, 20, 12, 30, 12, 28, 24, 24, 12, 50

Displayed as a triangular array:

2, 2,

2, 4, 2,

3, 4, 4, 3,

2, 6, 5, 6, 2,

4, 4, 6, 6, 4, 4,

2, 8, 4, 10, 4, 8, 2,

4, 4, 10, 6, 6, 10, 4, 4,

3, 8, 4, 12, 7, 12, 4, 8, 3,

MAPLE

with(numtheory):

T:= (n, k)-> add(add(phi(igcd(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_] := Outer[ EulerPhi[ GCD[#1, #2]]&, Divisors[n], Divisors[k]] // Flatten // Total; Table[ t[n-k+1, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 26 2013 *)

PROG

(Sage)

def A216620(n, k) :

cp = cartesian_product([divisors(n), divisors(k)])

return reduce(lambda x, y: x+y, map(euler_phi, map(gcd, cp)))

for n in (1..12): [A216620(n, k) for k in (1..12)]

CROSSREFS

Cf. A216621, A216622, A216623, A216624, A216625, A216626, A216627.

Sequence in context: A069930 A086327 A114896 * A181019 A066761 A108920

Adjacent sequences: A216617 A216618 A216619 * A216621 A216622 A216623

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Sep 12 2012

STATUS

approved