A212120 Triangle read by rows T(n,k), n>=1, k>=1, where T(n,k) is the sum of the divisors d of n with min(d, n/d) = k.

1, 3, 5, 7, 1, 9, 1, 11, 3, 13, 3, 15, 5, 17, 5, 1, 19, 7, 1, 21, 7, 1, 23, 9, 3, 25, 9, 3, 27, 11, 3, 29, 11, 5, 31, 13, 5, 1, 33, 13, 5, 1, 35, 15, 7, 1, 37, 15, 7, 1, 39, 17, 7, 3, 41, 17, 9, 3, 43, 19, 9, 3, 45, 19, 9, 3, 47, 21, 11, 5, 49, 21, 11, 5, 1

Offset: 1

Author

Omar E. Pol, Jul 02 2012

Keywords

• nonn
• tabf

Comments

Column k lists the odd numbers repeated k times starting in row k^2.

1 together with the first differences of the row sums give the divisor function A000005.

T(n,k) is also the total number of divisors of all positive integers <= n on the edges of k-th triangle in the diagram of divisors (see link section). See also A212119.

Examples

			Written as an irregular triangle the sequence begins:
1;
3;
5;
7,   1;
9,   1;
11,  3;
13,  3;
15,  5;
17,  5,  1;
19,  7,  1;
21,  7,  1;
23,  9,  3;
25,  9,  3;
27, 11,  3;
29, 11,  5;
31, 13,  5,  1;
33, 13,  5,  1;
35, 15,  7,  1;
37, 15,  7,  1;
39, 17,  7,  3;
41, 17,  9,  3;
43, 19,  9,  3;
45, 19,  9,  3;
47, 21, 11,  5;
49, 21, 11,  5,  1;
		

Links

• Omar E. Pol, Diagram of divisors, figure 1, figure 2.

Crossrefs

Row sums give A006218, n >= 1.

Columns (1-5): A005408, A109613, A130823, A129756, A130497.

Cf. A000005, A027750, A010766, A147861, A163100, A212119.

Formula

T(n,k) = Sum_{j=1..n} A212119(j,k).

Extensions

Definition changed by Franklin T. Adams-Watters, Jul 12 2012