login
A216621
Triangle read by rows, n >= 1, 1 <= k <= n, T(n,k) = Sum_{c|n,d|k} phi(gcd(c,d)).
8
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
1,2
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).
LINKS
EXAMPLE
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;
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 *)
PROG
(Sage)
for n in (1..9): [A216620(n, k) for k in (1..n)]
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Sep 12 2012
STATUS
approved