This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A384225 #32 Aug 22 2025 00:15:09 %S A384225 1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,1,1,1,3,1,1,2,1, %T A384225 1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,2,1,1,4,1,1,1,1,1,1,1,1,1,1,2,1,3, %U A384225 1,1,1,1,1,1,1,1,2,1,1,4,1,1,1,1,1,4,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4 %N A384225 Irregular triangle read by rows: T(n,k) is the number of odd divisors in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1. %C A384225 T(n,k) is the number of odd numbers in the k-th sublist of divisors of n whose terms increase by a factor of at most 2, %C A384225 In a 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 A384225 At least for the first 1000 rows the row lengths give A237271. %C A384225 Observation: at least the first 33 rows (or first 62 terms) coincide with A280940. %H A384225 Paolo Xausa, <a href="/A384225/b384225.txt">Table of n, a(n) for n = 1..10607</a> (rows 1..3500 of triangle, flattened). %e A384225 ------------------------------------------------------------------ %e A384225 | n | Row n of | List of divisors of n | Number of | %e A384225 | | the triangle | [with sublists in brackets] | sublists | %e A384225 ------------------------------------------------------------------ %e A384225 | 1 | 1; | [1]; | 1 | %e A384225 | 2 | 1; | [1, 2]; | 1 | %e A384225 | 3 | 1, 1; | [1], [3]; | 2 | %e A384225 | 4 | 1; | [1, 2, 4]; | 1 | %e A384225 | 5 | 1, 1; | [1], [5]; | 2 | %e A384225 | 6 | 2; | [1, 2, 3, 6]; | 1 | %e A384225 | 7 | 1, 1; | [1], [7]; | 2 | %e A384225 | 8 | 1; | [1, 2, 4, 8]; | 1 | %e A384225 | 9 | 1, 1, 1; | [1], [3], [9]; | 3 | %e A384225 | 10 | 1, 1; | [1, 2], [5, 10]; | 2 | %e A384225 | 11 | 1, 1; | [1], [11]; | 2 | %e A384225 | 12 | 2; | [1, 2, 3, 4, 6, 12]; | 1 | %e A384225 | 13 | 1, 1; | [1], [13]; | 2 | %e A384225 | 14 | 1, 1; | [1, 2], [7, 14]; | 2 | %e A384225 | 15 | 1, 2, 1; | [1], [3, 5], [15]; | 3 | %e A384225 | 16 | 1; | [1, 2, 4, 8, 16]; | 1 | %e A384225 | 17 | 1, 1; | [1], [17]; | 2 | %e A384225 | 18 | 3; | [1, 2, 3, 6, 9, 18]; | 1 | %e A384225 | 19 | 1, 1; | [1], [19]; | 2 | %e A384225 | 20 | 2; | [1, 2, 4, 5, 10, 20]; | 1 | %e A384225 | 21 | 1, 1, 1, 1; | [1], [3], [7], [21]; | 4 | %e A384225 ... %e A384225 For n = 14 the list of divisors of 14 is [1, 2, 7, 14]. There are two sublists of divisors of 14 whose terms increase by a factor of at most 2, they are [1, 2] and [7, 14]. Each sublist has only one odd number, so the row 14 is [1, 1]. %e A384225 For n = 15 the list of divisors of 15 is [1, 3, 5, 15]. There are three sublists of divisors of 15 whose terms increase by a factor of at most 2, they are [1], [3, 5], [15]. The number of odd numbers in the sublists are [1, 2, 1] respectively, so the row 15 is [1, 2, 1]. %e A384225 For n = 16 the list of divisors of 16 is [1, 2, 4, 8, 16]. There is only one sublist of divisors of 16 whose terms increase by a factor of at most 2, that is the same as the list of divisors of 16, which has five terms and only one odd number, so the row 16 is [1]. %t A384225 A384225row[n_] := Map[Count[#, _?OddQ] &, Split[Divisors[n], #2/# <= 2 &]]; %t A384225 Array[A384225row, 50] (* _Paolo Xausa_, Jul 08 2025 *) %Y A384225 Row sums give A001227. %Y A384225 Cf. A000203, A027750, A174973 (2-dense numbers), A280940, A237271, A379288, A384149, A384222, A384226, A384928, A384930, A384931, A385000, A386984, A386989, A387030. %K A384225 nonn,tabf,changed %O A384225 1,8 %A A384225 _Omar E. Pol_, Jun 16 2025