A140581 Triangle read by rows: T(n,k) = A140254(n/k) if k divides n, T(n,k) = 0 otherwise.

1, 1, 1, 2, 0, 1, 0, 1, 0, 1, 4, 0, 0, 0, 1, -3, 2, 1, 0, 0, 1, 6, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 1, -5, 4, 0, 0, 1, 0, 0, 0, 0, 1, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -3, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -7, 6, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1

Offset

1,4

Links

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

Formula

Mobius transform of triangle A140256 = A054525 * A140256 as infinite lower triangular matrices.

T(n,k) = Sum_{d|n} moebius(d)*A140256(n/d,k).

Example

Triangle begins:
   1;
   1, 1;
   2, 0, 1;
   0, 1, 0, 1;
   4, 0, 0, 0, 1;
  -3, 2, 1, 0, 0, 1;
   6, 0, 0, 0, 0, 0, 1;
   0, 0, 0, 1, 0, 0, 0, 1;
   0, 0, 2, 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, my(r=n/k); sumdiv(r, d, moebius(d)*b(r/d)), 0) \\ Andrew Howroyd, Sep 20 2025

Crossrefs

Column 1 is A140254.

Row sums are A014963.

Cf. A140256, A054525.

Sequence in context: A137276 A287234 A309938 * A137277 A387579 A039975

Adjacent sequences: A140578 A140579 A140580 * A140582 A140583 A140584

Keyword

tabl,sign

Author

Gary W. Adamson and Mats Granvik, May 17 2008

Extensions

Name edited and data corrected by Andrew Howroyd, Sep 20 2025

Status

approved