A379288 Irregular triangle read by rows in which row n lists the odd divisors of n except the "e" odd divisors described in A005279.

1, 1, 1, 3, 1, 1, 5, 1, 1, 7, 1, 1, 3, 9, 1, 5, 1, 11, 1, 1, 13, 1, 7, 1, 3, 15, 1, 1, 17, 1, 1, 19, 1, 1, 3, 7, 21, 1, 11, 1, 23, 1, 1, 5, 25, 1, 13, 1, 3, 9, 27, 1, 1, 29, 1, 1, 31, 1, 1, 3, 11, 33, 1, 17, 1, 5, 35, 1, 1, 37, 1, 19, 1, 3, 13, 39, 1, 1, 41, 1, 1, 43

Offset

1,4

Comments

Observation: row lengths coincide with at least the first 10000 terms of A237271.

It appears that the row lengths give A237271.

From Hartmut F. W. Hoft, Jan 09 2025: (Start)

Proof of the conjecture:

An entire part of SRS(n), n = 2^k * q with k >= 0 and q odd, up to the diagonal is described in row n of A249223 by a 1 in position d, an odd divisor of n, 0's in positions d-1 and 2^(k+1) * f, f >= d an odd divisor of n, and nonzero numbers that increase or decrease by 1 in between.

The odd divisors e of n with d < e < 2^(k+1) * f are the "e" odd divisors of A005279 since for divisor s of n, d < s < e < 2*s < 2^(k+1) * f holds.

The odd divisors u of n greater than A003056(n) are encoded by the 2^(k+1) * f above as u = q/f and odd divisors d < A003056(n) are also encoded as 2^(k+1) * q/d. Then odd divisors e of n with q/f < e < 2^(k+1) * q/d are the "e" odd divisors of A005279 since for divisor t of n, q/f < t < e < 2*t < 2^(k+1) * q/d holds.

For a part containing the diagonal the inequalities above hold on the respective sides of the diagonal.

As a consequence the number of entries in row n of this triangle equals A237271(n). (End)

Links

Table of n, a(n) for n=1..82.

Example

Triangle begins:
  1;
  1;
  1,  3;
  1;
  1,  5;
  1;
  1,  7;
  1:
  1,  3,  9;
  1,  5;
  1, 11;
  1;
  1, 13;
  1,  7;
  1,  3, 15;
  1;
  1, 17;
  1;
  1, 19;
  1;
  1,  3,  7, 21;
  ...

Mathematica

row[n_] := Module[{d = Partition[Divisors[n], 2, 1]}, Select[Join[{1}, Select[d, #[[2]] >= 2*#[[1]] &][[;; , 2]]], OddQ]]; Table[row[n], {n, 1, 50}] // Flatten (* Amiram Eldar, Dec 22 2024 *)

Crossrefs

These are the odd terms of A379374.

Subsequence of A182469.

Row sums give A379379.

Cf. A001227, A005279, A237270, A237271, A237593, A239657, A379384, A379461, A379634.

Cf. A000203, A003056, A249223.

Sequence in context: A016562 A087501 A294951 * A101443 A228037 A184726

Adjacent sequences: A379285 A379286 A379287 * A379289 A379290 A379291

Keyword

nonn,tabf

Author

Omar E. Pol, Dec 21 2024

Extensions

More terms from Amiram Eldar, Dec 22 2024

Status

approved