A127172 Triangle read by rows: A051731^3 as an infinite lower triangular matrix.

1, 3, 1, 3, 0, 1, 6, 3, 0, 1, 3, 0, 0, 0, 1, 9, 3, 3, 0, 0, 1, 3, 0, 0, 0, 0, 0, 1, 10, 6, 0, 3, 0, 0, 0, 1, 6, 0, 3, 0, 0, 0, 0, 0, 1, 9, 3, 0, 0, 3, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 18, 9, 6, 3, 0, 3, 0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset

1,2

Comments

As a linear operator T, this matrix applies the inverse Moebius transform three times. For example T * [1, 2, 3, ...] gives A007430.

Links

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

Formula

Equals A127170 * A051731 as infinite lower triangular matrices.

T(n,k) = A007425(n/k) if k divides n, T(n,k) = 0 otherwise.

Example

First few rows of the triangle:
   1;
   3, 1;
   3, 0, 1;
   6, 3, 0, 1;
   3, 0, 0, 0, 1;
   9, 3, 3, 0, 0, 1;
   3, 0, 0, 0, 0, 0, 1;
  10, 6, 0, 3, 0, 0, 0, 1;
   6, 0, 3, 0, 0, 0, 0, 0, 1;
   9, 3, 0, 0, 3, 0, 0, 0, 0, 1;
  ...

Mathematica

A127172[n_, k_] := If[Divisible[n, k], DivisorSum[n/k, DivisorSigma[0, #] &], 0];
Table[A127172[n, k], {n, 15}, {k, n}] (* Paolo Xausa, Sep 23 2025 *)

Prog

(PARI) T(n, k) = if(n%k, 0, sumdiv(n/k, d, numdiv(d))) \\ Andrew Howroyd, Sep 23 2025

Crossrefs

Row sums are A007426.

Column 1 is A007425

Cf. A007430, A051731, A127170 (square).

Sequence in context: A136157 A266260 A143353 * A011087 A180021 A091422

Adjacent sequences: A127169 A127170 A127171 * A127173 A127174 A127175

Keyword

nonn,tabl

Author

Gary W. Adamson, Jan 06 2007

Extensions

Corrected, edited and extended by Andrew Howroyd, Sep 23 2025

Status

approved