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 A361660 #46 Apr 13 2023 08:30:50 %S A361660 2,3,2,4,3,3,2,5,4,3,4,2,6,5,4,4,3,3,5,2,7,6,5,4,5,3,5,6,2,8,7,6,5,5, %T A361660 4,6,3,3,4,7,2,9,8,7,6,5,6,4,4,7,3,7,6,8,2,10,9,8,7,6,6,5,7,4,7,8,3,3, %U A361660 7,5,9,2,11,10,9,8,7,6,7,5,8,4,5,9,3,9,4,7,10,2 %N A361660 Irregular triangle read by rows where row n lists the successive numbers moved in the process of forming row n of the triangle A361642. %C A361660 The first and last numbers in row n>=2 are n and 2, respectively, and they occur just once each in the row. %C A361660 For row n>=3, and if and only if n-1 is prime, numbers n and 2 are the only numbers which occur just once (since when n-1 is prime it cannot make a rectangle for any other number to move from the initial column to the final row). %C A361660 A number can move twice in succession, and so occur here twice in succession, when it fills the top right corner cell in a rectangle of width * height = n. %C A361660 The move is from the initial column to top right corner cell, and therefore the numbers which appear twice in succession are d+1 for each divisor d of n, in the range 1 < d < n. %C A361660 If n is a prime, then it has no such divisors, or if n is a semiprime n = x*y (including square of a prime) then x+1 and y+1 are the only numbers appearing twice in succession. %C A361660 The length of row n is A002541(n). This equals to the number of special integer partitions of n there. Where a rectangle is formed of the changing shape, the row length increases more because the movement of a number that completes the rectangle is repeated as it continues to move again. %e A361660 The irregular triangle T(n,k) begins: %e A361660 n/k | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 %e A361660 ------------------------------------------------ %e A361660 1 | (empty row) %e A361660 2 | 2; %e A361660 3 | 3, 2; %e A361660 4 | 4, 3, 3, 2; %e A361660 5 | 5, 4, 3, 4, 2; %e A361660 6 | 6, 5, 4, 4, 3, 3, 5, 2; %e A361660 7 | 7, 6, 5, 4, 5, 3, 5, 6, 2; %e A361660 8 | 8, 7, 6, 5, 5, 4, 6, 3, 3, 4, 7, 2; %e A361660 9 | 9, 8, 7, 6, 5, 6, 4, 4, 7, 3, 7, 6, 8, 2; %e A361660 . %e A361660 Movements of the six-number-high column. 1 never moves. 4 and 3 move twice each in immediate succession as 6 is a composite and a semiprime: %e A361660 . %e A361660 6 %e A361660 5 5 %e A361660 4 4 4 %e A361660 3 3 3 3 4 3 %e A361660 2 2 2 5 2 5 2 5 2 5 3 2 5 2 %e A361660 1 1 6 1 6 1 6 1 6 4 1 6 4 1 6 4 3 1 6 4 3 5 1 6 4 3 5 2 %e A361660 . %e A361660 The parallel is shown for row length and the special integer partition in A002541: %e A361660 For n = 4, its row consists of 4, 3, 3 and 2, that is four elements. %e A361660 The special partition of n = 4 is (4), (2 2), (3 1), and (2 1 1), that is also four partitions. The relation is demonstrated by the illustration below. Square blocks represent the four numbers. As they move, the changing shape assumes a number of identical or reflected formations. The number of possible grouping of the blocks within them is exactly the same as the number of the moves that the blocks undergo: %e A361660 . _ _ %e A361660 | |__________ 1st move %e A361660 | | _ _ | %e A361660 | | | |_|____________ 2nd move ____________________________ 4th move %e A361660 | | | | | _ _ _v_ _|_ | %e A361660 | | | | | | | |____|___|_____ 3rd move | %e A361660 | | | |_v_ | | | | |_ _ _v_ _ _ _ _ _ _ _v_ %e A361660 | | | | | | | | | | | | | | %e A361660 |_ _| |_ _|_ _| |_ _|_ _| |_ _|_ _|_ _| |_ _ _ _ _ _ _ _| %e A361660 4 3 1 2 2 2 1 1 4 %e A361660 ^ ^ %e A361660 |____________________ Identical partition ________________| %Y A361660 Cf. A361642, A002541 (row length). %K A361660 nonn,tabf %O A361660 1,1 %A A361660 _Tamas Sandor Nagy_, Mar 19 2023