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

1,2

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.

Links

Paolo Xausa, Table of n, a(n) for n = 1..13282 (rows 1..3000 of triangle, flattened).

Example

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.

Mathematica

A384223row[n_] := Map[Total, Split[Divisors[n], EvenQ[#2] &]];
Array[A384223row, 50] (* Paolo Xausa, Sep 05 2025 *)

Prog

(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

Crossrefs

Row sums give A000203.

Row lengths give A001227.

Companion of A384224.

Cf. A000079, A027750.

Sequence in context: A209566 A208916 A209766 * A384149 A356207 A114972

Adjacent sequences: A384220 A384221 A384222 * A384224 A384225 A384226

Keyword

nonn,tabf,easy

Author

Omar E. Pol, Jun 03 2025

Status

approved