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%I A379288 #127 Aug 08 2025 00:05:50
%S A379288 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,
%T A379288 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,
%U A379288 33,1,17,1,5,35,1,1,37,1,19,1,3,13,39,1,1,41,1,1,43
%N A379288 Irregular triangle read by rows in which row n lists the odd divisors of n excluding odd divisors e for which there exists another divisor j with j < e < 2*j.
%C A379288 The excluded divisors are the odd divisors e listed in A005279.
%C A379288 Conjecture 1: the row lengths are given by A237271 (true for at least the first 10000 terms of A237271)
%C A379288 From _Hartmut F. W. Hoft_, Jan 09 2025: (Start)
%C A379288 Proof of Conjecture 1:
%C A379288 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.
%C A379288 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.
%C A379288 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.
%C A379288 For a part containing the diagonal the inequalities above hold on the respective sides of the diagonal.
%C A379288 As a consequence the number of entries in row n of this triangle equals A237271(n). (End)
%C A379288 From _Omar E. Pol_, Jun 26 2025: (Start)
%C A379288 Conjecture 2: T(n,m) is the smallest number in the m-th 2-dense sublist of divisors of n.
%C A379288 We call "2-dense sublists of divisors of n" to the maximal sublists of divisors of n whose terms increase by a factor of at most 2.
%C A379288 In a 2-dense sublist of divisors of n the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of n.
%C A379288 If the conjecture is true so row sums give A379379 and the row lengths give A237271, and the same row lengths have the sequences A384222, A384225 and A384226. Also the conjecture of A384149 should be true.
%C A379288 Observation: at least for the first 5000 rows (the first 15542 terms) the conjecture 2 coincides with the definition from the Name section and the row lengths give A237271.
%C A379288 An example of the conjecture 2, for n = 1..24 is as shown below:
%C A379288 -------------------------------------------------------------------
%C A379288 | n | Row n of | List of divisors of n | Number of |
%C A379288 | | the triangle | [with sublists in brackets] | sublists |
%C A379288 --------------------------------------------------------------------
%C A379288 | 1 | 1; | [1]; | 1 |
%C A379288 | 2 | 1; | [1, 2]; | 1 |
%C A379288 | 3 | 1, 3; | [1], [3]; | 2 |
%C A379288 | 4 | 1; | [1, 2, 4]; | 1 |
%C A379288 | 5 | 1, 5; | [1], [5]; | 2 |
%C A379288 | 6 | 1; | [1, 2, 3, 6]; | 1 |
%C A379288 | 7 | 1, 7; | [1], [7]; | 2 |
%C A379288 | 8 | 1; | [1, 2, 4, 8]; | 1 |
%C A379288 | 9 | 1, 3, 9; | [1], [3], [9]; | 3 |
%C A379288 | 10 | 1, 5; | [1, 2], [5, 10]; | 2 |
%C A379288 | 11 | 1, 11; | [1], [11]; | 2 |
%C A379288 | 12 | 1; | [1, 2, 3, 4, 6, 12]; | 1 |
%C A379288 | 13 | 1, 13; | [1], [13]; | 2 |
%C A379288 | 14 | 1, 7; | [1, 2], [7, 14]; | 2 |
%C A379288 | 15 | 1, 3, 15; | [1], [3, 5], [15]; | 3 |
%C A379288 | 16 | 1; | [1, 2, 4, 8, 16]; | 1 |
%C A379288 | 17 | 1, 17; | [1], [17]; | 2 |
%C A379288 | 18 | 1; | [1, 2, 3, 6, 9, 18]; | 1 |
%C A379288 | 19 | 1, 19; | [1], [19]; | 2 |
%C A379288 | 20 | 1; | [1, 2, 4, 5, 10, 20]; | 1 |
%C A379288 | 21 | 1, 3, 7, 21; | [1], [3], [7], [21]; | 4 |
%C A379288 | 22 | 1, 11; | [1, 2], [11, 22]; | 2 |
%C A379288 | 23 | 1, 23; | [1], [23]; | 2 |
%C A379288 | 24 | 1; | [1, 2, 3, 4, 6, 8, 12, 24]; | 1 |
%C A379288 ...
%C A379288 For n = 10 the list of divisors of 10 is [1, 2, 5, 10]. There are two 2-dense sublists of divisors of 10, they are [1, 2], [5, 10]. The smallest numbers in the sublists are [1, 5] respectively, so the row 10 is [1, 5].
%C A379288 For n = 15 the list of divisors of 15 is [1, 3, 5, 15]. There are three 2-dense sublists of divisors of 15, they are [1], [3, 5], [15]. The smallest numbers in the sublists are [1, 3, 15] respectively, so the row 15 is [1, 3, 15].
%C A379288 78 is the first practical number A005153 not in A174973. For n = 78 the list of divisors of 78 is [1, 2, 3, 6, 13, 26, 39, 78]. There are two 2-dense sublists of divisors of 78, they are [1, 2, 3, 6] and [13, 26, 39, 78]. The smallest numbers in the sublists are [1, 13] respectively, so the row 78 is [1, 13].
%C A379288 (End)
%C A379288 Conjecture 3: T(n,m) is the m-th divisor p of n such that p is greater than twice the adjacent previous divisor of n. - _Omar E. Pol_, Aug 02 2025
%t A379288 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 *)
%Y A379288 These are the odd terms of A379374.
%Y A379288 Subsequence of A182469.
%Y A379288 Row sums give A379379.
%Y A379288 Cf. A001227, A005279, A237270, A237271, A237593, A239657, A379384, A379461, A379634.
%Y A379288 Cf. A000203, A003056, A249223.
%Y A379288 Cf. A005153, A174973, A384149, A384222, A384225, A384226, A385000.
%K A379288 nonn,tabf
%O A379288 1,4
%A A379288 _Omar E. Pol_, Dec 21 2024
%E A379288 More terms from _Amiram Eldar_, Dec 22 2024