%I A143239
%S A143239 1,2,1,3,0,1,4,2,0,0,5,0,0,0,1,6,3,2,0,0,1,7,0,0,0,0,0,1,8,4,0,0,0,0,0,
               0,9,0,
%T A143239 3,0,0,0,0,0,0,10,5,0,0,2,0,0,0,0,1,11,0,0,0,0,0,0,0,0,0,1,12,6,4,0,0,
               2,0,0,0,0,
%U A143239 0,0,13,0,0,0,0,0,0,0,0,0,0,0,1,14,7,0,0,0,0,2,0,0,0,0,0,0,1
%V A143239 1,2,-1,3,0,-1,4,-2,0,0,5,0,0,0,-1,6,-3,-2,0,0,1,7,0,0,0,0,0,-1,8,-4,0,
               0,0,0,0,0,9,0,
%W A143239 -3,0,0,0,0,0,0,10,-5,0,0,-2,0,0,0,0,1,11,0,0,0,0,0,0,0,0,0,-1,12,-6,-4,
               0,0,2,0,0,0,0,
%X A143239 0,0,13,0,0,0,0,0,0,0,0,0,0,0,-1,14,-7,0,0,0,0,-2,0,0,0,0,0,0,1
%N A143239 Triangle read by rows, A126988 * A128407 as infinite lower triangular 
               matrices.
%C A143239 Row sums = A000010, phi(n): (1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4,...); 
               as a consequence of the Dedekind-Liouville rule illustrated in the 
               example and on p.137 of "Concrete Mathematics".
%D A143239 Ronald L. Graham, Donald E. Knuth & Oren Patashnik, "Concrete Mathematics" 
               2-nd ed.; Addison-Wesley, 1994, p. 137.
%F A143239 Triangle read by rows generated from the Dedekind-Liouville rule: T(n,
               k) = mu(k)*(n/k) if k divides n. T(n,k) = 0 if k is not a divisor 
               of n. Equals A126988 * A128407
%e A143239 First few rows of the triangle are:
%e A143239 1;
%e A143239 2, -1;
%e A143239 3, 0, -1;
%e A143239 4, -2, 0, 0;
%e A143239 5, 0, 0, 0, -1;
%e A143239 6, -3, -2, 0, 0, 1;
%e A143239 7, 0, 0, 0, 0, 0, -1;
%e A143239 8, -4, 0, 0, 0, 0, 0, 0;
%e A143239 9, 0, -3, 0, 0, 0, 0, 0, 0;
%e A143239 10, -5, 0, 0, -2, 0, 0, 0, 0, 1;
%e A143239 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1;
%e A143239 12, -6, -4, 0, 0, 2, 0, 0, 0, 0, 0, 0;
%e A143239 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1;
%e A143239 14, -7, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 1;
%e A143239 ...
%e A143239 Row 12 = (12, -6, -4, 0, 0, 2, 0, 0, 0, 0, 0, 0) since (Cf. A126988 - 
               the divisors of 12 are (12, 6, 4, 3, 0, 2, 0, 0, 0, 0, 0, 1) and 
               applying mu(k) * (nonzero terms), we get (1*12, (-1)*6, (-1)*4, 1*2) 
               sum = 4 = phi(12).
%Y A143239 Cf. A000010, A128407, A126988, A008683.
%Y A143239 Sequence in context: A141673 A127094 A158906 this_sequence A158951 A126988 
               A130026
%Y A143239 Adjacent sequences: A143236 A143237 A143238 this_sequence A143240 A143241 
               A143242
%K A143239 tabl,sign
%O A143239 1,2
%A A143239 Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 01 2008