A140699 Triangle read by rows: T(n,k) = moebius(k)*A014963(n/k) if k divides n, T(n,k) = 0 otherwise.

1, 2, -1, 3, 0, -1, 2, -2, 0, 0, 5, 0, 0, 0, -1, 1, -3, -2, 0, 0, 1, 7, 0, 0, 0, 0, 0, -1, 2, -2, 0, 0, 0, 0, 0, 0, 3, 0, -3, 0, 0, 0, 0, 0, 0, 1, -5, 0, 0, -2, 0, 0, 0, 0, 1, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, -1, -2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1

Offset

1,2

Comments

Similar to table A140256. Can perhaps be seen as taking the moebius function of A126988 times the mangoldt function of A126988.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Formula

Equals A054524 * A140256 as infinite lower triangular matrices.

A140700(n) = Product_{d|n} T(n,d).

Example

Triangle begins:
   1;
   2, -1;
   3,  0, -1;
   2, -2,  0, 0;
   5,  0,  0, 0, -1;
   1, -3, -2, 0,  0, 1;
   7,  0,  0, 0,  0, 0, -1;
   2, -2,  0, 0,  0, 0,  0, 0;
   3,  0, -3, 0,  0, 0,  0, 0, 0;
   1, -5,  0, 0, -2, 0,  0, 0, 0, 1;
  11,  0,  0, 0,  0, 0,  0, 0, 0, 0, -1;
  ...

Prog

(PARI) \\ here b(n) is A014963.
b(n)=ispower(n, , &n); if(isprime(n), n, 1)
T(n, k)=if(n%k, 0, moebius(k)*b(n/k)) \\ Andrew Howroyd, Sep 20 2025

Crossrefs

Row sums are A140254.

Column 1 is A014963.

Cf. A054524, A140256, A126988, A140700.

Sequence in context: A277322 A182740 A228786 * A140256 A126206 A307744

Adjacent sequences: A140696 A140697 A140698 * A140700 A140701 A140702

Keyword

sign,tabl

Author

Mats Granvik and Gary W. Adamson, May 24 2008

Extensions

New name from Andrew Howroyd, Sep 21 2025

Status

approved