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

Author

Gary W. Adamson, Aug 01 2008

Keywords

• nonn
• tabl
• easy

Comments

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

Examples

			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.
		

Links

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

Crossrefs

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

Programs

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

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 *)

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

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).