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%I #11 Sep 18 2017 03:09:32
%S 1,2,2,3,3,3,4,5,5,4,5,6,7,6,5,6,8,9,9,8,6,7,9,12,11,12,9,7,8,11,13,
%T 15,15,13,11,8,9,12,16,16,19,16,16,12,9,10,14,18,20,21,21,20,18,14,10,
%U 11,15,20,22,26,23,26,22,20,15,11,12,17,22,25,29,29,29,29,25,22,17,12
%N a(m, n) is the number of coprime pairs (i, j) with 1 <= i <= m, 1 <= j <= n; table of a(m, n) read by antidiagonals.
%C A kind of 2-dimensional version of the Euler phi function A000010.
%H Andrew Howroyd, <a href="/A135646/b135646.txt">Table of n, a(n) for n = 1..1275</a>
%F a(m, n) = Sum_{g=1..min(m,n)} floor(m/g) * floor(n/g) * moebius(g). - _Andrew Howroyd_, Sep 17 2017
%F a(n, n) = 2*(Sum_{i=1..n} phi(i)) - 1 = 2*A002088(n) - 1 = A018805(n).
%F a(m, n) <= m*n - Sum_{i=1..m} ( (i - phi(i)) * floor(n / i) ).
%F Conjecture: a(m, n) ~ mn - sum_1^m{ (i - phi(i)) (n / i) } = n sum_1^m{ phi(i) / i } ~ 6mn / pi^2 as m -> oo.
%F a(m, n) = A049687(m, n) + 2. - _Andrew Howroyd_, Sep 17 2017
%e a(2, 5) = 8 since of the 10 possible pairs all but (2, 2) and (2, 4) are coprime.
%e The terms given correspond to the following values:
%e 1 = a(1, 1)
%e 2 2 = a(2, 1), a(1, 2)
%e 3 3 3 = a(3, 1), a(2, 2), a(1, 3), etc.
%e 4 5 5 4
%e 5 6 7 6 5
%e 6 8 9 9 8 6
%e 7 9 12 11 12 9 7
%e 8 11 13 15 15 13 11 8
%e 9 12 16 16 19 16 16 12 9
%e 10 14 18 20 21 21 20 18 14 10
%e etc.
%o (PARI) a(m,n) = sum(g=1, min(m,n), (m\g)*(n\g)*moebius(g)); \\ _Andrew Howroyd_, Sep 17 2017
%Y Cf. A000010 (Euler's totient function), A002088 (sum of totient function), A018805.
%Y Cf. A049687.
%K nonn,tabl
%O 1,2
%A _Hugo van der Sanden_, Nov 22 2008
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