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A112593 Triangle where a(1,1) = 1, a(n,m) = number of terms of row (n-1) which are coprime to m. Row n has (2n-1) terms.

%I A112593 #15 Oct 11 2019 16:44:48
%S A112593 1,1,1,1,3,3,3,3,3,5,5,0,5,5,0,5,7,5,5,5,0,5,5,5,5,9,8,8,8,1,8,7,8,8,
%T A112593 1,8,11,4,10,4,11,3,10,4,10,4,11,3,11,13,6,11,6,10,4,13,6,11,6,9,4,13,
%U A112593 6,8,15,6,9,6,14,5,15,6,9,6,13,5,12,6,8,6,15,17,8,5,8,12,3,16,8,5,3,17,3,16
%N A112593 Triangle where a(1,1) = 1, a(n,m) = number of terms of row (n-1) which are coprime to m. Row n has (2n-1) terms.
%C A112593 GCD(m,0) is considered here to be m, so 0 is coprime to no positive integer but 1.
%e A112593 Row 5 of the triangle is [7,5,5,5,0,5,5,5,5].
%e A112593 Among these terms there are 9 terms coprime to 1, 8 terms coprime to 2, 8 terms coprime to 3, 8 terms coprime to 4, 1 term coprime to 5, 8 terms coprime to 6, 7 terms coprime to 7, 8 terms coprime to 8, 8 terms coprime to 9, 1 term coprime to 10 and 8 terms coprime to 11. So row 6 is [9,8,8,8,1,8,7,8,8,1,8].
%e A112593 Table begins:
%e A112593 1,
%e A112593 1,1,1,
%e A112593 3,3,3,3,3,
%e A112593 5,5,0,5,5,0,5,
%e A112593 7,5,5,5,0,5,5,5,5,
%e A112593 9,8,8,8,1,8,7,8,8,1,8,
%e A112593 11,4,10,4,11,3,10,4,10,4,11,3,11,
%e A112593 13,6,11,6,10,4,13,6,11,6,9,4,13,6,8,
%e A112593 15,6,9,6,14,5,15,6,9,6,13,5,12,6,8,6,15,
%e A112593 17,8,5,8,12,3,16,8,5,3,17,3,16,8,3,8,17,3,17
%t A112593 f[l_] := Append[l, Table[ Count[GCD[Last[l], n], 1], {n, Length[Last[l]] + 2}]]; Flatten[Nest[f, {{1}}, 9]] (* _Ray Chandler_, Jan 02 2006 *)
%t A112593 t[1, 1] = 1; t[n_, m_] := t[n, m] = Count[ GCD[ Table[ t[n - 1, k], {k, 2n - 3}], m], 1]; Table[ t[n, m], {n, 10}, {m, 2n - 1}] // Flatten (* _Robert G. Wilson v_ *)
%o A112593 (PARI) {print1(s=1,",");v=[s];for(i=2,10,w=vector(2*i-1);for(j=1,2*i-1,c=0;for(k=1,2*i-3,if(gcd(v[k],j)==1,c++));print1(w[j]=c,","));v=w)} (Brockhaus)
%Y A112593 Cf. A112599.
%Y A112593 Row sums are in A160991. [From _Klaus Brockhaus_, Jun 01 2009]
%K A112593 nonn,tabf
%O A112593 1,5
%A A112593 _Leroy Quet_, Dec 24 2005
%E A112593 More terms from _Robert G. Wilson v_, _Klaus Brockhaus_ and _Ray Chandler_, Jan 02 2006