A134577 Triangle read by rows: T(n,k) = tau(n/k)*k if k divides n, T(n,k) = 0 otherwise, where tau = A000005.

1, 2, 2, 2, 0, 3, 3, 4, 0, 4, 2, 0, 0, 0, 5, 4, 4, 6, 0, 0, 6, 2, 0, 0, 0, 0, 0, 7, 4, 6, 0, 8, 0, 0, 0, 8, 3, 0, 6, 0, 0, 0, 0, 0, 9, 4, 4, 0, 0, 10, 0, 0, 0, 0, 10, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 6, 8, 9, 8, 0, 12, 0, 0, 0, 0, 0, 12, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13

Offset

1,2

Comments

As an infinite lower triangular matrix T, some known sequence transforms:

T * [1/1, 1/2, 1/3, ...] = A007425;

T * [1, 2, 3, ...] = A007433;

T * A000005 = A034761.

In general, T * X applies the inverse Moebius transform twice to the termwise product of the vector X and the natural numbers. - Andrew Howroyd, Sep 23 2025

Links

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

Formula

Equals A127170 * A127648 = A051731^2 * A127648 = A051731 * A127093 as infinite lower triangular matrices.

T(n,k) = k*A127170(n,k) = Sum_{d|n, k|d} k. - Andrew Howroyd, Sep 23 2025

Example

First few rows of the triangle:
  1;
  2, 2;
  2, 0, 3;
  3, 4, 0, 4;
  2, 0, 0, 0, 5
  4, 4, 6, 0, 0, 6;
  2, 0, 0, 0, 0, 0, 7;
  4, 6, 0, 8, 0, 0, 0, 8;
  ...

Mathematica

A134577[n_, k_] := If[Divisible[n, k], DivisorSigma[0, n/k]*k, 0];
Table[A134577[n, k], {n, 15}, {k, n}] (* Paolo Xausa, Sep 23 2025 *)

Prog

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

Crossrefs

Row sums are A007429.

Column 1 is A000005.

Cf. A051731, A127170, A127648, A127093, A007429, A127170, A007433, A034761.

Sequence in context: A071443 A013586 A071441 * A071442 A174740 A124759

Adjacent sequences: A134574 A134575 A134576 * A134578 A134579 A134580

Keyword

nonn,tabl

Author

Gary W. Adamson, Nov 02 2007

Extensions

New name and terms a(56) onwards from Andrew Howroyd, Sep 23 2025

Status

approved