A143728 Triangle read by rows: termwise product of mu(n) and n-th row of A127368.

1, 1, 0, 1, -2, 0, 1, 0, -3, 0, 1, -2, -3, 0, 0, 1, 0, 0, 0, -5, 0, 1, -2, -3, 0, -5, 6, 0, 1, 0, -3, 0, -5, 0, -7, 0, 1, -2, 0, 0, -5, 0, -7, 0, 0, 1, 0, -3, 0, 0, 0, -7, 0, 0, 0, 1, -2, -3, 0, -5, 6, -7, 0, 0, 10, 0, 1, 0, 0, 0, -5, 0, -7, 0, 0, 0, -11, 0, 1, -2, -3, 0, -5, 6, -7, 0, 0, 10, -11, 0, 0

Offset: 1

Author

Gary W. Adamson, Aug 30 2008

Keywords

• tabl
• sign

Comments

The operation A127368 * A128407 forms the termwise product of mu(n) and the n-th row of A127368: deleting all squares and changing the sign of primes to (-1).

Row sums = A143729: (1, 1, -1, -2, -4, -4, -3, -14, ...)

Examples

			First few terms of the triangle:
  1;
  1,  0;
  1, -2,  0;
  1,  0, -3,  0;
  1, -2, -3,  0,  0;
  1,  0,  0,  0, -5,  0;
  1, -2, -3,  0, -5,  6,  0;
  1,  0, -3,  0, -5,  0, -7,  0;
  ...
Example: row 7 = (1, -2, -3, 0, -5, 6, 0). We take row 7 of triangle A127368 which records the relative primes of 7 as (1, 2, 3, 4, 5, 6, 0). Applying the termwise product of the first 7 terms of mu(k): (1, -1, -1, 0, -1, 1, -1), we get (1, -2, -3, 0, -5, 6, 0), noting that the "4" has been deleted.
		

Crossrefs

Cf. A008683, A127368, A128407.

Formula

Triangle read by rows, A127368 * A128407, 1 <= k <= n; T(n,k) = {1<=k<=n, gcd(k,n)=1} * mu(k).

Extensions

Partially edited by N. J. A. Sloane, Jan 05 2009

a(66) = 0 inserted by Georg Fischer, Jun 05 2023