

A135646


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.


2



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, 15, 15, 13, 11, 8, 9, 12, 16, 16, 19, 16, 16, 12, 9, 10, 14, 18, 20, 21, 21, 20, 18, 14, 10, 11, 15, 20, 22, 26, 23, 26, 22, 20, 15, 11, 12, 17, 22, 25, 29, 29, 29, 29, 25, 22, 17, 12
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OFFSET

1,2


COMMENTS

A kind of 2dimensional version of the Euler phi function A000010.


LINKS



FORMULA

a(m, n) = Sum_{g=1..min(m,n)} floor(m/g) * floor(n/g) * moebius(g).  Andrew Howroyd, Sep 17 2017
a(n, n) = 2*(Sum_{i=1..n} phi(i))  1 = 2*A002088(n)  1 = A018805(n).
a(m, n) <= m*n  Sum_{i=1..m} ( (i  phi(i)) * floor(n / i) ).
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.


EXAMPLE

a(2, 5) = 8 since of the 10 possible pairs all but (2, 2) and (2, 4) are coprime.
The terms given correspond to the following values:
1 = a(1, 1)
2 2 = a(2, 1), a(1, 2)
3 3 3 = a(3, 1), a(2, 2), a(1, 3), etc.
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 15 15 13 11 8
9 12 16 16 19 16 16 12 9
10 14 18 20 21 21 20 18 14 10
etc.


PROG

(PARI) a(m, n) = sum(g=1, min(m, n), (m\g)*(n\g)*moebius(g)); \\ Andrew Howroyd, Sep 17 2017


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



