A331781 - OEIS

%I #20 Dec 04 2022 19:07:58

%S 0,0,1,0,2,3,0,3,5,7,0,4,6,9,11,0,5,8,12,15,19,0,6,9,13,16,21,23,0,7,

%T 11,16,20,26,29,35,0,8,12,18,22,29,32,39,43,0,9,14,20,25,33,36,44,49,

%U 55,0,10,15,22,27,35,38,47,52,59,63,0,11,17,25,31,40,44,54,60,68,73,83

%N Triangle read by rows: T(m,n) = Sum_{0<i<m, 0<j<n, gcd{i,j}=1} 1, where m >= n >= 1.

%H M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. <a href="https://doi.org/10.1137/140978090">On the minimal teaching sets of two-dimensional threshold functions</a>. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090. See Lemma 11.

%e Triangle begins:

%e 0,

%e 0, 1,

%e 0, 2, 3,

%e 0, 3, 5, 7,

%e 0, 4, 6, 9, 11,

%e 0, 5, 8, 12, 15, 19,

%e 0, 6, 9, 13, 16, 21, 23,

%e 0, 7, 11, 16, 20, 26, 29, 35,

%e 0, 8, 12, 18, 22, 29, 32, 39, 43,

%e 0, 9, 14, 20, 25, 33, 36, 44, 49, 55

%e ...

%p VS := proc(m,n) local a,i,j; a:=0;

%p for i from 1 to m-1 do for j from 1 to n-1 do

%p if gcd(i,j)=1 then a:=a+1; fi; od: od: a; end;

%p for m from 1 to 12 do lprint([seq(VS(m,n),n=1..m)]); od:

%t Table[Sum[Boole[# == 1] # &@ GCD[i, j], {i, m - 1}, {j, n - 1}], {m, 12}, {n, m}] // Flatten (* _Michael De Vlieger_, Feb 12 2020 *)

%Y Main diagonal is A018805.

%Y A333295 is essentially the same array.

%K nonn,tabl

%O 1,5

%A _N. J. A. Sloane_, Feb 11 2020