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 A379630 #76 Dec 30 2024 17:02:30 %S A379630 1,1,2,1,3,1,3,1,4,1,5,2,5,1,6,1,3,1,7,3,7,1,8,1,9,4,2,3,9,1,10,1,5,1, %T A379630 11,5,11,1,12,3,3,1,13,6,13,1,14,2,7,1,15,7,4,1,5,3,15,1,16,1,17,8,17, %U A379630 1,18,5,3,9,3,1,19,9,19,1,20,2,5,1,21,10,6,1,7,3,21,1,22,4,11,1,23,11,23,1,24,7,3,1 %N A379630 Irregular triangle read by rows in which row n lists the smallest parts of the partitions of n into consecutive parts followed by the conjugate corresponding odd divisors of n in accordance with the theorem of correspondence described in the Comments lines. %C A379630 Theorem of correspondence between the partitions of n into k consecutive parts and the odd divisors of n: given a partition of n into k consecutive parts if k is odd then the corresponding odd divisor of n is k, otherwise if k is even then the corresponding odd divisor of n is the sum of any pair of conjugate parts of the partition (for example the sum of the largest part and the smallest part). %C A379630 Conjecture: the first A001227(n) terms in the n-th row are also the absolute values of the n-th row of A341971. %C A379630 The last A001227(n) terms in the n-th row are also the mirror of the n-th row of A261697. %e A379630 Triangle begins: %e A379630 1, 1; %e A379630 2, 1; %e A379630 3, 1, 3, 1; %e A379630 4, 1; %e A379630 5, 2, 5, 1; %e A379630 6, 1, 3, 1; %e A379630 7, 3, 7, 1; %e A379630 8, 1; %e A379630 9, 4, 2, 3, 9, 1; %e A379630 10, 1, 5, 1; %e A379630 11, 5, 11, 1; %e A379630 12, 3, 3, 1; %e A379630 13, 6, 13, 1; %e A379630 14, 2, 7, 1; %e A379630 15, 7, 4, 1, 5, 3, 15, 1; %e A379630 16, 1; %e A379630 17, 8, 17, 1; %e A379630 18, 5, 3, 9, 3, 1; %e A379630 19, 9, 19, 1; %e A379630 20, 2, 5, 1; %e A379630 21, 10, 6, 1, 7, 3, 21, 1; %e A379630 ... %e A379630 For n = 21 the partitions of 21 into consecutive parts are [21], [11, 10], [8, 7, 6], [6, 5, 4, 3, 2, 1]. %e A379630 On the other hand the odd divisors of 21 are [1, 3, 7, 21]. %e A379630 To determine how these partitions are related to the odd divisors we follow the two rules of the theorem as shown below: %e A379630 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 A379630 The second partition is [11, 10] and the number of parts is 2 and 2 even so the corresponding odd divisor of 21 is equal to 11 + 10 = 21. %e A379630 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 A379630 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 A379630 Summarizing in a table: %e A379630 -------------------------------------- %e A379630 Correspondence %e A379630 -------------------------------------- %e A379630 Partitions of 21 Odd %e A379630 into consecutive divisors %e A379630 parts of 21 %e A379630 ------------------- ---------- %e A379630 [21] .................... 1 %e A379630 [11, 10] ................ 21 %e A379630 [8, 7, 6] ................ 3 %e A379630 [6, 5, 4, 3, 2, 1] ....... 7 %e A379630 . %e A379630 Then we can make a table of conjugate correspondence in which the four partitions are arrenged in four columns with the smallest parts at the top as shown below: %e A379630 ------------------------------------------ %e A379630 Conjugate correspondence %e A379630 ------------------------------------------ %e A379630 Partitions of 21 Odd %e A379630 into consecutive divisors %e A379630 parts as columns of 21 %e A379630 ------------------- ------------------ %e A379630 21 10 6 1 7 3 21 1 %e A379630 | 11 7 2 | | | | %e A379630 | | 8 3 | | | | %e A379630 | | | 4 | | | | %e A379630 | | | 5 | | | | %e A379630 | | | 6 | | | | %e A379630 | | | |_______| | | | %e A379630 | | |_________________| | | %e A379630 | |___________________________| | %e A379630 |_____________________________________| %e A379630 . %e A379630 Then removing all rows except the first row we have a table of conjugate correspondence for smallest parts and odd divisors as shown below: %e A379630 ------------------- ------------------ %e A379630 Smallest parts Odd divisors %e A379630 ------------------- ------------------ %e A379630 21 10 6 1 7 3 21 1 %e A379630 | | | |_______| | | | %e A379630 | | |_________________| | | %e A379630 | |___________________________| | %e A379630 |_____________________________________| %e A379630 . %e A379630 So the 21st row of the triangle is [21, 10, 6, 1, 7, 3, 21, 1]. %e A379630 . %e A379630 Illustration of initial terms in an isosceles triangle demonstrating the theorem: %e A379630 . _ _ %e A379630 _|1|1|_ %e A379630 _|2 _|_ 1|_ %e A379630 _|3 |1|3| 1|_ %e A379630 _|4 _| | |_ 1|_ %e A379630 _|5 |2 _|_ 5| 1|_ %e A379630 _|6 _| |1|3| |_ 1|_ %e A379630 _|7 |3 | | | 7| 1|_ %e A379630 _|8 _| _| | |_ |_ 1|_ %e A379630 _|9 |4 |2 _|_ 3| 9| 1|_ %e A379630 _|10 _| | |1|5| | |_ 1|_ %e A379630 _|11 |5 _| | | | |_ 11| 1|_ %e A379630 _|12 _| |3 | | | 3| |_ 1|_ %e A379630 _|13 |6 | _| | |_ | 13| 1|_ %e A379630 _|14 _| _| |2 _|_ 7| |_ |_ 1|_ %e A379630 _|15 |7 |4 | |1|5| | 3| 15| 1|_ %e A379630 _|16 _| | | | | | | | |_ 1|_ %e A379630 _|17 |8 _| _| | | | |_ |_ 17| 1|_ %e A379630 _|18 _| |5 |3 | | | 9| 3| |_ 1|_ %e A379630 _|19 |9 | | _| | |_ | | 19| 1|_ %e A379630 _|20 _| _| | |2 _|_ 5| | |_ |_ 1|_ %e A379630 |21 |10 |6 | | |1|7| | | 3| 21| 1| %e A379630 . %e A379630 The geometrical structure of the above isosceles triangle is defined in A237593. See also the triangle A286001. %e A379630 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 A379630 . %Y A379630 Column 1 gives A000027. %Y A379630 Right border gives A000012. %Y A379630 The sum of row n equals A286014(n) + A000593(n). %Y A379630 The length of row n is A054844(n) = 2*A001227(n). %Y A379630 The partitions of n into consecutive parts are in the n-th row of A299765. See also A286000 and A286001. %Y A379630 The odd divisors of n are in the n-th row of A182469. See also A261697 and A261699. %Y A379630 Cf. A196020, A204217, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A245092, A262626, A341971. %K A379630 nonn,tabf %O A379630 1,3 %A A379630 _Omar E. Pol_, Dec 28 2024