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 A379631 #33 Jan 07 2025 10:51:53 %S A379631 1,1,2,1,3,2,3,1,4,1,5,3,5,1,6,3,3,1,7,4,7,1,8,1,9,5,4,3,9,1,10,4,5,1, %T A379631 11,6,11,1,12,5,3,1,13,7,13,1,14,5,7,1,15,8,6,5,5,3,15,1,16,1,17,9,17, %U A379631 1,18,7,6,9,3,1,19,10,19,1,20,6,5,1,21,11,8,6,7,3,21,1,22,7,11,1,23,12,23,1,24,9,3,1 %N A379631 Irregular triangle read by rows: T(n,m), n >= 1, m >= 1, in which row n lists the largest parts of the partitions of n into consecutive parts followed by the conjugate corresponding odd divisors of n in accordance with the theorem described in A379630. %C A379631 Consider that the mentioned partitions are ordered by increasing number of parts. %C A379631 Row n gives the n-th row of A379633 together with the n-th row of A379634. %e A379631 Triangle begins: %e A379631 1, 1; %e A379631 2, 1; %e A379631 3, 2, 3, 1; %e A379631 4, 1; %e A379631 5, 3, 5, 1; %e A379631 6, 3, 3, 1; %e A379631 7, 4, 7, 1; %e A379631 8, 1; %e A379631 9, 5, 4, 3, 9, 1, %e A379631 10, 4, 5, 1; %e A379631 11, 6, 11, 1; %e A379631 12, 5, 3, 1; %e A379631 13, 7, 13, 1; %e A379631 14, 5, 7, 1; %e A379631 15, 8, 6, 5, 5, 3, 15, 1; %e A379631 16, 1; %e A379631 17, 9, 17, 1; %e A379631 18, 7, 6, 9, 3, 1; %e A379631 19, 10, 19, 1; %e A379631 20, 6, 5, 1; %e A379631 21, 11, 8, 6, 7, 3, 21, 1; %e A379631 ... %e A379631 For n = 21 the partitions of 21 into consecutive parts are [21], [11, 10], [8, 7, 6], [6, 5, 4, 3, 2, 1]. %e A379631 On the other hand the odd divisors of 21 are [1, 3, 7, 21]. %e A379631 To determine how these partitions are related to the odd divisors we follow the two rules of the theorem described in A379630 as shown below: %e A379631 The first partition is [21] and the number of parts is 1 and 1 is odd so the corresponding odd divisor of 21 is 1. %e A379631 The second partition is [11, 10] and the number of parts is 2 and 2 is even so the corresponding odd divisor of 21 is equal to 11 + 10 = 21. %e A379631 The third partition is [8, 7, 6] and the number of parts is 3 and 3 is odd so the corresponding odd divisor of 21 is 3. %e A379631 The fourth partition is [6, 5, 4, 3, 2, 1] and the number of parts is 6 and 6 is even so the corresponding odd divisor of 21 is equal to 6 + 1 = 5 + 2 = 4 + 3 = 7. %e A379631 Summarizing in a table: %e A379631 -------------------------------------- %e A379631 Correspondence %e A379631 -------------------------------------- %e A379631 Partitions of 21 Odd %e A379631 into consecutive divisors %e A379631 parts of 21 %e A379631 ------------------- ---------- %e A379631 [21] .................... 1 %e A379631 [11, 10] ................ 21 %e A379631 [8, 7, 6] ................ 3 %e A379631 [6, 5, 4, 3, 2, 1] ....... 7 %e A379631 . %e A379631 Then we can make a table of conjugate correspondence in which the four partitions are arrenged in four columns with the largest parts at the top as shown below: %e A379631 ------------------------------------------ %e A379631 Conjugate correspondence %e A379631 ------------------------------------------ %e A379631 Partitions of 21 Odd %e A379631 into consecutive divisors %e A379631 parts as columns of 21 %e A379631 ------------------- ------------------ %e A379631 21 11 8 6 7 3 21 1 %e A379631 | 10 7 5 | | | | %e A379631 | | 6 4 | | | | %e A379631 | | | 3 | | | | %e A379631 | | | 2 | | | | %e A379631 | | | 1 | | | | %e A379631 | | | |_______| | | | %e A379631 | | |_________________| | | %e A379631 | |___________________________| | %e A379631 |_____________________________________| %e A379631 . %e A379631 Then removing all rows except the first row we have a table of conjugate correspondence for largest parts and odd divisors as shown below: %e A379631 ------------------- ------------------ %e A379631 Largest parts Odd divisors %e A379631 ------------------- ------------------ %e A379631 21 11 8 6 7 3 21 1 %e A379631 | | | |_______| | | | %e A379631 | | |_________________| | | %e A379631 | |___________________________| | %e A379631 |_____________________________________| %e A379631 . %e A379631 So the 21st row of the triangle is [21, 11, 8, 6, 7, 3, 21, 1]. %e A379631 . %e A379631 Illustration of initial terms in an isosceles triangle demonstrating the theorem described in A379630: %e A379631 . _ _ %e A379631 _|1|1|_ %e A379631 _|2 _|_ 1|_ %e A379631 _|3 |2|3| 1|_ %e A379631 _|4 _| | |_ 1|_ %e A379631 _|5 |3 _|_ 5| 1|_ %e A379631 _|6 _| |3|3| |_ 1|_ %e A379631 _|7 |4 | | | 7| 1|_ %e A379631 _|8 _| _| | |_ |_ 1|_ %e A379631 _|9 |5 |4 _|_ 3| 9| 1|_ %e A379631 _|10 _| | |4|5| | |_ 1|_ %e A379631 _|11 |6 _| | | | |_ 11| 1|_ %e A379631 _|12 _| |5 | | | 3| |_ 1|_ %e A379631 _|13 |7 | _| | |_ | 13| 1|_ %e A379631 _|14 _| _| |5 _|_ 7| |_ |_ 1|_ %e A379631 _|15 |8 |6 | |5|5| | 3| 15| 1|_ %e A379631 _|16 _| | | | | | | | |_ 1|_ %e A379631 _|17 |9 _| _| | | | |_ |_ 17| 1|_ %e A379631 _|18 _| |7 |6 | | | 9| 3| |_ 1|_ %e A379631 _|19 |10 | | _| | |_ | | 19| 1|_ %e A379631 _|20 _| _| | |6 _|_ 5| | |_ |_ 1|_ %e A379631 |21 |11 |8 | | |6|7| | | 3| 21| 1| %e A379631 . %e A379631 The geometrical structure of the above isosceles triangle is defined in A237593. See also the triangles A286000 and A379633. %e A379631 Note that the diagram also can be interpreted as a template which after folding gives a 90 degree pop-up card which has essentially the same structure as the stepped pyramid described in A245092. %e A379631 . %Y A379631 Column 1 gives A000027. %Y A379631 Right border gives A000012. %Y A379631 The sum of row n equals A286015(n) + A000593(n). %Y A379631 The length of row n is A054844(n) = 2*A001227(n). %Y A379631 For another version with smallest parts see A379630. %Y A379631 The partitions of n into consecutive parts are in the n-th row of A299765. See also A286000. %Y A379631 The odd divisors of n are in the n-th row of A182469. See also A261697 and A261699. %Y A379631 Cf. A196020, A204217, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A245092, A262626, A379632, A379633, A379634. %K A379631 nonn,tabf %O A379631 1,3 %A A379631 _Omar E. Pol_, Dec 30 2024