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

1,2

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).

Links

Table of n, a(n) for n=1..75.

Index entries for sequences related to sigma(n)

Example

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.

Prog

(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

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.

Sequence in context: A121845 A239639 A099816 * A272025 A262411 A280488

Adjacent sequences: A237270 A237271 A237272 * A237274 A237275 A237276

Keyword

nonn,tabf

Author

Omar E. Pol, Feb 08 2014

Status

approved