%I #18 Apr 01 2017 04:58:31

%S 1,2,4,5,10,7,18,16,23,19,42,24,58,38,46,56,96,52,120,72,93,93,172,91,

%T 171,132,176,143,270,116,308,218,237,228,280,201,432,286,330,275,530,

%U 237,584,368,394,417,696,357,666,431,570,515,882,452,716,565,712,665

%N a(n) = Sum_{k=1..n} (number of positive integers <= k which are coprime to both k and n).

%C Equals row sums of triangle A144379. - _Gary W. Adamson_, Sep 19 2008

%H Giovanni Resta, <a href="/A125728/b125728.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{j=1..n} Sum_{k|(n*j)} mu(k) * floor(j/k), where mu(k) is the Mobius (Moebius) function and the inner sum is over the positive divisors, k, of (n*j).

%e The positive integers coprime to k and <= k are, as k runs from 1 to 8, 1; 1; 1, 2; 1,3; 1,2,3,4; 1,5; 1,2,3,4,5,6; 1,3,5,7. So we want, so as to get a(8), the number of 1's, 3's, 5's and 7's in this concatenated list, since the positive integers <=8 and coprime to 8 are 1,3,5,7. In the concatenated list there are eight 1's, four 3's, three 5's and one 7. So a(8) = 8 + 4 + 3 + 1 = 16.

%t f[n_] := Sum[Sum[ Boole[GCD[j, k] == 1 && GCD[j, n] == 1], {j, k}], {k, n}];Table[f[n], {n, 60}] (* _Ray Chandler_, Feb 03 2007 *)

%Y Cf. A144379. - _Gary W. Adamson_, Sep 19 2008

%K nonn

%O 1,2

%A _Leroy Quet_, Feb 02 2007

%E Extended by _Ray Chandler_, Feb 03 2007