A236540 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists k copies of the positive squares in nondecreasing order, except the first column which lists the triangular numbers, and the first element of column k is in row k(k+1)/2.

0, 1, 3, 1, 6, 1, 10, 4, 15, 4, 1, 21, 9, 1, 28, 9, 1, 36, 16, 4, 45, 16, 4, 1, 55, 25, 4, 1, 66, 25, 9, 1, 78, 36, 9, 1, 91, 36, 9, 4, 105, 49, 16, 4, 1, 120, 49, 16, 4, 1, 136, 64, 16, 4, 1, 153, 64, 25, 9, 1, 171, 81, 25, 9, 1, 190, 81, 25, 9, 4, 210, 100, 36, 9, 4, 1

Offset: 1

Author

Omar E. Pol, Jan 28 2014

Keywords

• nonn
• tabf

Comments

Gives an identity for the sum of all aliquot divisors of all positive integers <= n.

Alternating sum of row n equals A153485(n), i.e., Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = A153485(n).

Row n has length A003056(n) hence the first element of column k is in row A000217(k).

Column 1 is A000217. Columns 2-3 are A008794, A211547, but without the zeros.

Column k lists the partial sums of the k-th column of triangle A231347 which gives an identity for the sum of aliquot divisors of n. - Omar E. Pol, Nov 11 2014

Examples

			Triangle begins:
    0;
    1;
    3,   1;
    6,   1;
   10,   4;
   15,   4,   1;
   21,   9,   1;
   28,   9,   1;
   36,  16,   4;
   45,  16,   4,   1;
   55,  25,   4,   1;
   66,  25,   9,   1;
   78,  36,   9,   1;
   91,  36,   9,   4;
  105,  49,  16,   4,  1;
  120,  49,  16,   4,  1;
  136,  64,  16,   4,  1;
  153,  64,  25,   9,  1;
  171,  81,  25,   9,  1;
  190,  81,  25,   9,  4;
  210, 100,  36,   9,  4,  1;
  231, 100,  36,  16,  4,  1;
  253, 121,  36,  16,  4,  1;
  276, 121,  49,  16,  4,  1;
  ...
For n = 6 the divisors of all positive integers <= 6 are [1], [1, 2], [1, 3], [1, 2, 4], [1, 5], [1, 2, 3, 6] hence the sum of all aliquot divisors is [0] + [1] + [1] + [1+2] + [1] + [1+2+3] = 0 + 1 + 1 + 3 + 1 + 6 = 12. On the other hand the 6th row of triangle is 15, 4, 1, therefore the alternating row sum is 15 - 4 + 1 = 12, equaling the sum of all aliquot divisors of all positive integers <= 6.
		

Crossrefs

Cf. A000203, A000217, A001065, A008794, A003056, A153485, A196020, A211547, A211343, A228813, A231345, A231347, A235791, A235794, A235799, A236104, A236106, A236112, A237591, A237593, A286001.