A228812 Triangle read by rows: T(n,k), n>=1, k>=1, in which row n lists m terms, where m = A055086(n). If k divides n and k < n^(1/2) then T(n,k) = k and T(n,m-k+1) = n/T(n,k). Also, if k^2 = n then T(n,k) = k. Other terms are zeros.

1, 1, 2, 1, 3, 1, 2, 4, 1, 0, 5, 1, 2, 3, 6, 1, 0, 0, 7, 1, 2, 4, 8, 1, 0, 3, 0, 9, 1, 2, 0, 5, 10, 1, 0, 0, 0, 11, 1, 2, 3, 4, 6, 12, 1, 0, 0, 0, 0, 13, 1, 2, 0, 0, 7, 14, 1, 0, 3, 5, 0, 15, 1, 2, 0, 4, 0, 8, 16, 1, 0, 0, 0, 0, 0, 17, 1, 2, 3, 0, 6, 9, 18

Offset: 1

Author

Omar E. Pol, Oct 03 2013

Keywords

• nonn
• tabf

Comments

The number of positive terms of row n is A000005(n).

The positive terms of row n are the divisors of n in increasing order.

Row n has length A055086(n).

Column k starts in row A002620(k+1).

The number of zeros in row n equals A078152(n).

The sum of row n is A000203(n).

Positive terms give A027750.

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.

For another version see A228814.

Examples

			For n = 60 the 60th row of triangle is [1, 2, 3, 4, 5, 6, 0, 0, 10, 12, 15, 20, 30, 60]. The row length is A055086(60) = 14. The number of zeros is A078152(60) = 2. The number of positive terms is A000005(60) = 12. The positive terms are the divisors of 60. The row sum is A000203(60) = 168.
Triangle begins:
1;
1,  2;
1,  3;
1,  2,  4;
1,  0,  5;
1,  2,  3,  6;
1,  0,  0,  7;
1,  2,  4,  8;
1,  0,  3,  0,  9;
1,  2,  0,  5, 10;
1,  0,  0,  0, 11;
1,  2,  3,  4,  6, 12;
1,  0,  0,  0,  0, 13;
1,  2,  0,  0,  7, 14;
1,  0,  3,  5,  0, 15;
1,  2,  0,  4,  0,  8, 16;
1,  0,  0,  0,  0,  0, 17;
1,  2,  3,  0,  6,  9, 18;
1,  0,  0,  0,  0,  0, 19;
1,  2,  0,  4,  5,  0, 10, 20;
1,  0,  3,  0,  0,  7,  0, 21;
1,  2,  0,  0,  0,  0, 11, 22;
1,  0,  0,  0,  0,  0,  0, 23;
1,  2,  3,  4,  6,  8, 12, 24;
...
		

Crossrefs

Column 1 is A000012.

Right border gives A000027.

Cf. A000005, A000203, A002620, A004526, A018253, A027750, A055086, A078152, A127093, A147861, A161904, A196020, A210959, A212119, A212120, A221645, A228813, A228814.