A231347 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the odd numbers interleaved with k-1 zeros but T(n,1) = n - 1 and the first element of column k is in row k(k+1)/2.

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

Offset: 1

Author

Omar E. Pol, Dec 28 2013

Keywords

• nonn
• tabf

Comments

Alternating sum of row n equals the sum of aliquot divisors of n, i.e., sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = A001065(n).

Row n has length A003056(n).

Column k starts in row A000217(k).

The number of positive terms in row n is A001227(n), for n >= 2.

If n = 2^j then the only positive integer in row n is T(n,1) = n - 1, for j >= 1.

If n is an odd prime then the only two positive integers in row n are T(n,1) = n - 1 and T(n,2) = n - 2.

Examples

			Triangle begins:
  0;
  1;
  2,   1;
  3,   0;
  4,   3;
  5,   0,  1;
  6,   5,  0;
  7,   0,  0;
  8,   7,  3;
  9,   0,  0,  1;
  10,  9,  0,  0;
  11,  0,  5,  0;
  12, 11,  0,  0;
  13,  0,  0,  3;
  14, 13,  7,  0,  1;
  15,  0,  0,  0,  0;
  16, 15,  0,  0,  0;
  17,  0,  9,  5,  0;
  18, 17,  0,  0,  0;
  19,  0,  0,  0,  3;
  20, 19, 11,  0,  0,  1;
  21,  0,  0,  7,  0,  0;
  22, 21,  0,  0,  0,  0;
  23,  0, 13,  0,  0,  0;
  ...
For n = 15 the aliquot divisors of 15 are 1, 3, 5, therefore the sum of aliquot divisors of 15 is 1 + 3 + 5 = 9. On the other hand the 15th row of triangle is 14, 13, 7, 0, 1, hence the alternating row sum is 14 - 13 + 7 - 0 + 1 = 9, equalling the sum of aliquot divisors of 15.
If n is even then the alternating sum of the n-th row of triangle is simpler than the sum of aliquot divisors of n. Example: the sum of aliquot divisors of 24 is 1 + 2 + 3 + 4 + 6 + 8 + 12 = 36, and the alternating sum of the 24th row of triangle is 23 - 0 + 13 - 0 + 0 - 0 = 36.
		

Crossrefs

Columns 1-2: A001477, A193356.

Cf. A000217, A001065, A001227, A000203, A003056, A196020, A211343, A212119, A228813, A231345, A235791, A235794, A236104, A236106, A236112, A237591, A237593, A286001.

Formula

T(n,1) = n - 1.

T(n,k) = A196020(n,k), for k >= 2.