cp's OEIS Frontend

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

%I A216622 #19 Mar 08 2020 00:06:30
%S A216622 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,
%T A216622 14,14,20,20,14,14,8,9,16,21,24,13,24,21,16,9,10,18,24,28,30,30,28,24,
%U A216622 18,10,11,20,19,26,35,28,35,26,19,20,11,12,22,30,36,40
%N A216622 Square array read by antidiagonals: T(n,k) = Sum_{c|n, d|k} phi(lcm(c,d)) for n >= 1, k >= 1.
%C A216622 T(n,n) = A062380(n) = Sum_{d|n} phi(d)*tau(d^2).
%C A216622 T(n,1) = T(1,n) = A000027(n) = n.
%C A216622 T(n,2) = T(2,n) = A005843(n) = 2*n.
%C A216622 T(n+1,n) = A002378(n) = (n+1)*n.
%C A216622 T(prime(n),1) = A000040(n) = prime(n).
%C A216622 T(prime(n),prime(n)) = 3*prime(n)-2.
%H A216622 Alois P. Heinz, <a href="/A216622/b216622.txt">Antidiagonals n = 1..141, flattened</a>
%e A216622 [-----1---2---3---4---5---6---7---8---9---10---11---12]
%e A216622 [ 1]  1,  2,  3,  4,  5,  6,  7,  8,  9,  10,  11,  12
%e A216622 [ 2]  2,  4,  6,  8, 10, 12, 14, 16, 18,  20,  22,  24
%e A216622 [ 3]  3,  6,  7, 12, 15, 14, 21, 24, 19,  30,  33,  28
%e A216622 [ 4]  4,  8, 12, 14, 20, 24, 28, 26, 36,  40,  44,  42
%e A216622 [ 5]  5, 10, 15, 20, 13, 30, 35, 40, 45,  26,  55,  60
%e A216622 [ 6]  6, 12, 14, 24, 30, 28, 42, 48, 38,  60,  66,  56
%e A216622 [ 7]  7, 14, 21, 28, 35, 42, 19, 56, 63,  70,  77,  84
%e A216622 [ 8]  8, 16, 24, 26, 40, 48, 56, 42, 72,  80,  88,  78
%e A216622 [ 9]  9, 18, 19, 36, 45, 38, 63, 72, 37,  90,  99,  76
%e A216622 [10] 10, 20, 30, 40, 26, 60, 70, 80, 90,  52, 110, 120
%e A216622 [11] 11, 22, 33, 44, 55, 66, 77, 88, 99, 110,  31, 132
%e A216622 [12] 12, 24, 28, 42, 60, 56, 84, 78, 76, 120, 132,  98
%e A216622 .
%e A216622 Displayed as a triangular array:
%e A216622    1,
%e A216622    2,  2,
%e A216622    3,  4,  3,
%e A216622    4,  6,  6,  4,
%e A216622    5,  8,  7,  8,  5,
%e A216622    6, 10, 12, 12, 10,  6,
%e A216622    7, 12, 15, 14, 15, 12,  7,
%e A216622    8, 14, 14, 20, 20, 14, 14,  8,
%e A216622    9, 16, 21, 24, 13, 24, 21, 16,  9,
%p A216622 with(numtheory):
%p A216622 T:= (n, k)-> add(add(phi(ilcm(c, d)), c=divisors(n)), d=divisors(k)):
%p A216622 seq (seq (T(n, 1+d-n), n=1..d), d=1..14);  # _Alois P. Heinz_, Sep 12 2012
%t A216622 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 *)
%o A216622 (Sage)
%o A216622 def A216622(n, k) :
%o A216622     cp = cartesian_product([divisors(n), divisors(k)])
%o A216622     return reduce(lambda x,y: x+y, map(euler_phi, map(lcm, cp)))
%o A216622 for n in (1..12): [A216622(n,k) for k in (1..12)]
%Y A216622 Cf. A216620, A216621, A216623, A216624, A216625, A216626, A216627.
%K A216622 nonn,tabl
%O A216622 1,2
%A A216622 _Peter Luschny_, Sep 12 2012