A384223 Irregular triangle read by rows: T(n,k) is the sum of the k-th odd divisor and the next even divisors that are less than the next odd divisor of n, with n >= 1, k >= 1.

1, 3, 1, 3, 7, 1, 5, 3, 9, 1, 7, 15, 1, 3, 9, 3, 15, 1, 11, 3, 25, 1, 13, 3, 21, 1, 3, 5, 15, 31, 1, 17, 3, 9, 27, 1, 19, 7, 35, 1, 3, 7, 21, 3, 33, 1, 23, 3, 57, 1, 5, 25, 3, 39, 1, 3, 9, 27, 7, 49, 1, 29, 3, 3, 21, 45, 1, 31, 63, 1, 3, 11, 33, 3, 51, 1, 5, 7, 35, 3, 13, 75, 1, 37, 3, 57, 1, 3, 13, 39, 7, 83

Offset: 1

Author

Omar E. Pol, Jun 03 2025

Keywords

• nonn
• tabf
• easy

Comments

If n is odd the row n lists the divisors of n.

If n is a power of 2 then row n is 2*n - 1.

Examples

			Triangle begins:
   1;
   3;
   1,  3;
   7;
   1,  5;
   3,  9;
   1,  7;
  15;
   1,  3,  9;
   3, 15;
   1, 11;
   3, 25;
   1, 13;
   3, 21;
   1,  3,  5,  15;
  31;
  ...
For n = 30 the list of divisors of 30 is [1, 2, 3, 5, 6, 10, 15, 30]. There are four sublists of divisors whose first term is odd. They are [1, 2], [3], [5, 6, 10], [15, 30]. The sum of the divisors in the sublists are respectively [3, 3, 21, 45], the same as the 30th row of the triangle.
		

Crossrefs

Row sums give A000203.

Row lengths give A001227.

Companion of A384224.

Cf. A000079, A027750, A237270, A237271, A237593, A279387, A384149.

Programs

PARI

row(n) = {my(d = divisors(n), res = vector(#d / (valuation(n, 2)+1)), t = 1, s = 1); for(i = 2, #d, if(bitand(d[i], 1), res[t] = s; t++; s = d[i], s += d[i])); res[#res] = s; res} \\ David A. Corneth, Jun 08 2025