A134542 Triangle read by rows: T(n,k) = Sum_{i=k..n} A134541(n,i).

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 3, 2, 1, 1, 2, 3, 3, 2, 1, 1, 3, 4, 4, 3, 2, 1, 1, 3, 4, 4, 4, 3, 2, 1, 1, 3, 4, 5, 5, 4, 3, 2, 1, 1, 2, 4, 5, 5, 5, 4, 3, 2, 1, 1, 3, 5, 6, 6, 6, 5, 4, 3, 2, 1, 1, 3, 4, 5, 6, 6, 6, 5, 4, 3, 2, 1, 1, 4, 5, 6, 7, 7, 7, 6, 5, 4, 3, 2, 1

Offset

1,5

Links

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

Formula

Equals A134541 * A000012 as infinite lower triangular matrices.

A015631(n) = Sum_{k=1..n} k*T(n,k).

T(n,k) = 1 - Sum_{j=1..k-1} A002321(n/j). - Andrew Howroyd, Sep 20 2025

Example

First few rows of the triangle:
  1;
  1, 1;
  1, 2, 1;
  1, 2, 2, 1;
  1, 3, 3, 2, 1;
  1, 2, 3, 3, 2, 1;
  1, 3, 4, 4, 3, 2, 1;
  1, 3, 4, 4, 4, 3, 2, 1;
  1, 3, 4, 5, 5, 4, 3, 2, 1;
  1, 2, 4, 5, 5, 5, 4, 3, 2, 1;
  ...

Prog

(PARI) \\ here b(n) is A002321.
b(n) = sum(i=1, n, moebius(i))
T(n, k) = sum(i=k, n, b(n\i)) \\ Andrew Howroyd, Sep 20 2025

Crossrefs

Row sums are A002088.

Cf. A002321, A015631, A134541, A134543.

Sequence in context: A301608 A285521 A187451 * A106254 A117147 A111007

Adjacent sequences: A134539 A134540 A134541 * A134543 A134544 A134545

Keyword

nonn,tabl

Author

Gary W. Adamson, Oct 31 2007

Extensions

New name and a(56) onwards from Andrew Howroyd, Sep 20 2025

Status

approved