A212119 Triangle read by rows T(n,k), n>=1, k>=1, where T(n,k) is the number of divisors d of n with min(d, n/d) = k.

1, 2, 2, 2, 1, 2, 0, 2, 2, 2, 0, 2, 2, 2, 0, 1, 2, 2, 0, 2, 0, 0, 2, 2, 2, 2, 0, 0, 2, 2, 0, 2, 0, 2, 2, 2, 0, 1, 2, 0, 0, 0, 2, 2, 2, 0, 2, 0, 0, 0, 2, 2, 0, 2, 2, 0, 2, 0, 2, 2, 0, 0, 2, 0, 0, 0, 2, 2, 2, 2, 2, 0, 0, 0, 1, 2, 2, 0, 0, 0, 2, 0, 2, 0, 0

Offset: 1

Author

Omar E. Pol, Jul 02 2012

Keywords

• nonn
• tabf

Comments

Column k lists the numbers A040000: 1, 2, 2, 2, 2... interleaved with k-1 zeros, starting in row k^2.

The sum of row n gives A000005(n), the number of divisors of n.

T(n,k) is also the number of divisors of n on the edges of k-th triangle in the diagram of divisors (see link section). See also A212120.

It appears that there are only eight rows that do not contain zeros. The indices of these rows are 1, 2, 3, 4, 6, 8, 12, 24, the divisors of 24, see A018253. - Omar E. Pol, Dec 03 2013

Examples

			Row 10 gives 2, 2, 0 therefore the sums of row 10 is 2+2+0 = 4, the same as A000005(10), the number of divisors of 10.
Written as an irregular triangle the sequence begins:
1;
2;
2;
2, 1;
2, 0;
2, 2;
2, 0;
2, 2;
2, 0, 1;
2, 2, 0;
2, 0, 0;
2, 2, 2;
2, 0, 0;
2, 2, 0;
2, 0, 2;
2, 2, 0, 1;
2, 0, 0, 0;
2, 2, 2, 0;
2, 0, 0, 0;
2, 2, 0, 2;
2, 0, 2, 0;
2, 2, 0, 0;
2, 0, 0, 0;
2, 2, 2, 2;
2, 0, 0, 0, 1;
		

Links

• Omar E. Pol, Diagram of divisors, figure 1, figure 2
• Omar E. Pol, Illustration of initial terms and of row sums

Crossrefs

Row sums give A000005. Column 1 is A040000. Column 2 gives the absolute values of A176742.

Cf. A006218, A027750, A010766, A147861, A163100, A196020, A210959, A212120, A211343, A221645, A228812-A228814.

Extensions

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