A237273 Triangle read by rows: T(n,k) = k+m, if k < m and k*m = n, or T(n,k) = k, if k^2 = n. Otherwise T(n,k) = 0. With n>=1 and 1<=k<=A000196(n).

1, 3, 4, 5, 2, 6, 0, 7, 5, 8, 0, 9, 6, 10, 0, 3, 11, 7, 0, 12, 0, 0, 13, 8, 7, 14, 0, 0, 15, 9, 0, 16, 0, 8, 17, 10, 0, 4, 18, 0, 0, 0, 19, 11, 9, 0, 20, 0, 0, 0, 21, 12, 0, 9, 22, 0, 10, 0, 23, 13, 0, 0, 24, 0, 0, 0, 25, 14, 11, 10, 26, 0, 0, 0, 5

Offset: 1

Author

Omar E. Pol, Feb 08 2014

Keywords

• nonn
• tabf

Comments

The first element of column k is in row k^2.

Column k lists k, k-1 zeros, and the positive integers but starting from 2*k+1 interleaved with k-1 zeros.

Row n has only one positive term iff n is a noncomposite number (A008578).

It appears that there are only eight rows that do not contain zeros. The indices of these rows are in A018253 (the divisors of 24).

Examples

			Triangle begins:
1;
3;
4;
5,   2;
6,   0;
7,   5;
8,   0;
9,   6;
10,  0,  3;
11,  7,  0;
12,  0,  0;
13,  8,  7;
14,  0,  0;
15,  9,  0;
16,  0,  8;
17, 10,  0,  4;
18,  0,  0,  0;
19, 11,  9,  0;
20,  0,  0,  0;
21, 12,  0,  9;
22,  0, 10,  0;
23, 13,  0,  0;
24,  0,  0,  0;
25, 14, 11, 10;
26,  0,  0,  0,   5;
27, 15,  0,  0,   0;
28,  0, 12,  0,   0;
29, 16,  0, 11,   0;
30,  0,  0,  0,   0;
31, 17, 13,  0,  11;
...
For n = 9 the divisors of n are 1, 3, 9, so row 9 is 10, 0, 3, because 1*9 = 9 and 3^2 = 9. The sum of row 9 is A000203(9) = 13.
For n = 12 the divisors of 12 are 1, 2, 3, 4, 6, 12, so row 12 is 13, 8, 7, because 1*12 = 12, 2*6 = 12 and 3*4 = 12. The sum of row 12 is A000203(12) = 28.
		

Links

• Index entries for sequences related to sigma(n)

Crossrefs

Row sums give A000203.

Row n has length A000196(n).

Column 1 is A065475.

Cf. A000290, A008578, A018253, A027750, A196020, A210959, A212119, A212120, A228812-A228814, A231347, A236104, A236631, A237519, A237593.

Programs

PARI

T(n, k) = if (n % k, 0, if (k^2==n, k, k + n/k));
tabf(nn) = {for (n = 1, nn, v = vector(sqrtint(n), k, T(n, k)); print(v););} \\ Michel Marcus, Jun 19 2019