A110661 Triangle read by rows: T(n,k) = total number of divisors of k, k+1, ..., n (1 <= k <= n).

1, 3, 2, 5, 4, 2, 8, 7, 5, 3, 10, 9, 7, 5, 2, 14, 13, 11, 9, 6, 4, 16, 15, 13, 11, 8, 6, 2, 20, 19, 17, 15, 12, 10, 6, 4, 23, 22, 20, 18, 15, 13, 9, 7, 3, 27, 26, 24, 22, 19, 17, 13, 11, 7, 4, 29, 28, 26, 24, 21, 19, 15, 13, 9, 6, 2, 35, 34, 32, 30, 27, 25, 21, 19, 15, 12, 8, 6, 37, 36, 34

Offset: 1

Author

Emeric Deutsch, Aug 02 2005

Keywords

• nonn
• tabl

Comments

Equals A000012 * (A000005 * 0^(n-k)) * A000012, 1 <= k <= n. - Gary W. Adamson, Jul 26 2008

Row sums = A143127. - Gary W. Adamson, Jul 26 2008

Examples

			T(4,2)=7 because 2 has 2 divisors, 3 has 2 divisors and 4 has 3 divisors.
Triangle begins:
   1;
   3, 2;
   5, 4, 2;
   8, 7, 5, 3;
  10, 9, 7, 5, 2;
  ...
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

Cf. A000005, A006218.

Cf. A143127.

Programs

Maple

with(numtheory): T:=(n,k)->add(tau(j),j=k..n): for n from 1 to 13 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form

Mathematica

T[n_, n_] := DivisorSigma[0, n]; T[n_, k_] := Sum[DivisorSigma[0, j], {j, k, n}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* G. C. Greubel, Sep 03 2017 *)

Formula

T(n, k) = Sum_{j=k..n} tau(j), where tau(j) is the number of divisors of j, and 1 <= k <= n.

T(n,n) = tau(n) = A000005(n) = number of divisors of n.

T(n,1) = Sum_{j=1..n} tau(j) = A006218(n).