A143235 Triangle read by rows: T(n,k) = tau(n)*tau(k), the product of the number of divisors.

1, 2, 4, 2, 4, 4, 3, 6, 6, 9, 2, 4, 4, 6, 4, 4, 8, 8, 12, 8, 16, 2, 4, 4, 6, 4, 8, 4, 4, 8, 8, 12, 8, 16, 8, 16, 3, 6, 6, 9, 6, 12, 6, 12, 9, 4, 8, 8, 12, 8, 16, 8, 16, 12, 16, 2, 4, 4, 6, 4, 8, 4, 8, 6, 8, 4, 6, 12, 12, 18, 12, 24, 12, 24, 18, 24, 12, 36, 2, 4, 4, 6, 4, 8, 4, 8, 6, 8, 4, 12, 4

Offset

1,2

Comments

The triangle can also be created by the triple matrix product A130209 * A000012 * A130209.

Links

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Formula

T(n,k) = A000005(n)*A000005(k), for 1 <= k <= n, n >= 1.

Sum_{k=1..n} T(n, k) = A143236(n) (row sums).

Example

First few rows of the triangle =
  1;
  2, 4;
  2, 4, 4;
  3, 6, 6,  9;
  2, 4, 4,  6, 4;
  4, 8, 8, 12, 8, 16;
  2, 4, 4,  6, 4,  8, 4;
  4, 8, 8, 12, 8, 16, 8, 16;
  3, 6, 6,  9, 6, 12, 6, 12, 9;
  ...
T(9,6) = 12 = d(9)*d(6) = 3*4.

Mathematica

A143235[n_, k_]:= DivisorSigma[0, n]*DivisorSigma[0, k];
Table[A143235[n, k], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Sep 12 2024 *)

Prog

(Magma)
A143235:= func< n, k | NumberOfDivisors(n)*NumberOfDivisors(k) >;
[A143235(n, k): k in [1..n], n in [1..14]]; // G. C. Greubel, Sep 12 2024
(SageMath)
def A143235(n, k): return sigma(n, 0)*sigma(k, 0)
flatten([[A143235(n, k) for k in range(1, n+1)] for n in range(1, 15)]) # G. C. Greubel, Sep 12 2024

Crossrefs

Cf. A000005, A035116 (right diagonal), A143236 (row sums).

Sequence in context: A004020 A363290 A246436 * A069465 A047947 A018838

Adjacent sequences: A143232 A143233 A143234 * A143236 A143237 A143238

Keyword

nonn,tabl,easy

Author

Gary W. Adamson, Aug 01 2008

Status

approved