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 A320051 #33 Sep 03 2023 09:36:13 %S A320051 3,5,1,7,2,6,10,4,12,72,11,8,15,144,120,13,9,20,288,180,1800,14,16,24, %T A320051 400,240,3528,840,17,18,28,450,252,4050,1080,3600,19,25,30,576,336, %U A320051 5184,1260,7200,2520,21,32,35,648,360,7056,1440,14112,5040,28800,22,36,40,800,378,8100,1680,14400,5544 %N A320051 Square array read by antidiagonals upwards: T(n,k) is the n-th positive integer with exactly k middle divisors, n >= 1, k >= 0. %C A320051 This is a permutation of the natural numbers. %C A320051 For the definition of middle divisors see A067742. %C A320051 Conjecture 1: T(n,k) is also the n-th positive integer j with the property that the difference between the number of partitions of j into an odd number of consecutive parts and the number of partitions of j into an even number of consecutive parts is equal to k. %C A320051 Conjecture 2: T(n,k) is also the n-th positive integer j with the property that the symmetric representation of sigma(j) has width k on the main diagonal. %H A320051 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a> %e A320051 The corner of the square array begins: %e A320051 3, 1, 6, 72, 120, 1800, 840, 3600, 2520, 28800, ... %e A320051 5, 2, 12, 144, 180, 3528, 1080, 7200, 5040, ... %e A320051 7, 4, 15, 288, 240, 4050, 1260, 14112, ... %e A320051 10, 8, 20, 400, 252, 5184, 1440, ... %e A320051 11, 9, 24, 450, 336, 7056, ... %e A320051 13, 16, 28, 576, 360, ... %e A320051 14, 18, 30, 648, ... %e A320051 17, 25, 35, ... %e A320051 19, 32, ... %e A320051 21, ... %e A320051 ... %e A320051 In accordance with the conjecture 1, T(1,0) = 3 because there is only one partition of 3 into an odd number of consecutive parts: [3], and there is only one partition of 3 into an even number of consecutive parts: [2, 1], therefore the difference of the number of those partitions is 1 - 1 = 0. %e A320051 On the other hand, in accordance with the conjecture 2: T(1,0) = 3 because the symmetric representation of sigma(3) = 4 has width 0 on the main diagonal, as shown below: %e A320051 . _ _ %e A320051 . |_ _|_ %e A320051 . | | %e A320051 . |_| %e A320051 . %e A320051 In accordance with the conjecture 1, T(1,2) = 6 because there are three partitions of 6 into an odd number of consecutive parts: [6], [3, 2, 1], and there are no partitions of 6 into an even number of consecutive parts, therefore the difference of the number of those partitions is 2 - 0 = 2. %e A320051 On the other hand, in accordance with the conjecture 2: T(1,2) = 6 because the symmetric representation of sigma(6) = 12 has width 2 on the main diagonal, as shown below: %e A320051 . _ _ _ _ %e A320051 . |_ _ _ |_ %e A320051 . | |_ %e A320051 . |_ _ | %e A320051 . | | %e A320051 . | | %e A320051 . |_| %e A320051 . %Y A320051 Row 1 is A128605. %Y A320051 Column 0 is A071561. %Y A320051 The union of the rest of the columns gives A071562. %Y A320051 Column 1 is A320137. %Y A320051 Column 2 is A320142. %Y A320051 For more information about the diagrams see A237593. %Y A320051 For tables of partitions into consecutive parts see A286000 and A286001. %Y A320051 Cf. A067742, A240542, A245092, A249351 (widths), A262626, A279286, A280849, A281007, A299761, A299777, A303297, A319529, A319796, A319801, A319802. %K A320051 nonn,tabl %O A320051 1,1 %A A320051 _Omar E. Pol_, Oct 04 2018