OFFSET
1,3
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.
FORMULA
T(n,1) = n - 1.
T(n,k) = A196020(n,k), for k >= 2.
EXAMPLE
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
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Dec 28 2013
STATUS
approved