A110662 Triangle read by rows: T(n,k) is the sum of the sums of divisors of k, k+1, ..., n (1 <= k <= n).

1, 4, 3, 8, 7, 4, 15, 14, 11, 7, 21, 20, 17, 13, 6, 33, 32, 29, 25, 18, 12, 41, 40, 37, 33, 26, 20, 8, 56, 55, 52, 48, 41, 35, 23, 15, 69, 68, 65, 61, 54, 48, 36, 28, 13, 87, 86, 83, 79, 72, 66, 54, 46, 31, 18, 99, 98, 95, 91, 84, 78, 66, 58, 43, 30, 12, 127, 126, 123, 119, 112

Offset: 1

Author

Emeric Deutsch, Aug 02 2005

Keywords

• nonn
• tabl

Comments

Equals A000012 * (A000203 * 0^(n-k)) * A000012, 1 <= k <= n. - Gary W. Adamson, Jul 26 2008

Row sums = A143128. - Gary W. Adamson, Jul 26 2008

Examples

			T(4,2)=14 because the divisors of 2 are {1,2}, the divisors of 3 are {1,3} and the divisors of 4 are {1,2,4}; sum of all these divisors is 14.
Triangle begins:
   1;
   4,  3;
   8,  7,  4;
  15, 14, 11,  7;
  21, 20, 17, 13,  6;
  ...
		

Links

• Indranil Ghosh, Rows 1..100, flattened

Crossrefs

Cf. A000203, A024916.

Cf. A143128.

Programs

Maple

with(numtheory): T:=(n,k)->add(sigma(j),j=k..n): for n from 1 to 12 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form

Mathematica

T[n_, n_] := DivisorSigma[1, n]; T[n_, k_] := Sum[DivisorSigma[1, 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} sigma(j), where sigma(j) is the sum of the divisors of j.

T(n, n) = sigma(n) = A000203(n) = sum of divisors of n.

T(n, 1) = Sum_{j=1..n} sigma(j) = A024916(n).