1,2

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".

Ronald L. Graham, Donald E. Knuth & Oren Patashnik, "Concrete Mathematics" 2-nd ed.; Addison-Wesley, 1994, p. 137.

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

First few rows of the triangle are:

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, -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, 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;

...

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).

Cf. A000010, A128407, A126988, A008683.

Sequence in context: A141673 A127094 A158906 this_sequence A158951 A126988 A130026

Adjacent sequences: A143236 A143237 A143238 this_sequence A143240 A143241 A143242

tabl,sign

Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 01 2008