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 A346871 #49 Aug 26 2021 16:41:30 %S A346871 2,2,1,3,2,4,2,1,6,3,1,1,7,3,2,1,9,4,2,1,1,10,4,2,2,1,12,5,2,2,1,1,15, %T A346871 6,3,2,1,1,1,16,6,3,2,2,1,1,19,7,4,2,2,1,1,1,21,8,4,2,2,2,1,1,22,8,4, %U A346871 3,2,1,2,1,24,9,4,3,2,2,1,1,1,27,10,5,3,2,2,1,2,1 %N A346871 Irregular triangle read by rows in which row n lists the row A000040(n) of A237591, n >= 1. %C A346871 The characteristic shape of the symmetric representation of sigma(prime(n)) consists in that the diagram contains exactly two regions (or parts) and each region is a rectangle (or bar), except for the first prime number (the 2) whose symmetric representation of sigma(2) consists of only one region which contains three cells. %C A346871 So knowing this characteristic shape we can know if a number is prime (or not) just by looking at the diagram, even ignoring the concept of prime number. %C A346871 Therefore we can see a geometric pattern of the exact distribution of prime numbers in the stepped pyramid described in A245092. %C A346871 T(n,k) is the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(prime(n)), from the border to the center, hence the sum of the n-th row of triangle is equal to A000040(n). %C A346871 T(n,k) is also the difference between the total number of partitions of all positive integers <= n-th prime into exactly k consecutive parts, and the total number of partitions of all positive integers <= n-th prime into exactly k + 1 consecutive parts. %e A346871 Triangle begins: %e A346871 2; %e A346871 2, 1; %e A346871 3, 2; %e A346871 4, 2, 1; %e A346871 6, 3, 1, 1; %e A346871 7, 3, 2, 1; %e A346871 9, 4, 2, 1, 1; %e A346871 10, 4, 2, 2, 1; %e A346871 12, 5, 2, 2, 1, 1; %e A346871 15, 6, 3, 2, 1, 1, 1; %e A346871 16, 6, 3, 2, 2, 1, 1; %e A346871 19, 7, 4, 2, 2, 1, 1, 1; %e A346871 21, 8, 4, 2, 2, 2, 1, 1; %e A346871 22, 8, 4, 3, 2, 1, 2, 1; %e A346871 24, 9, 4, 3, 2, 2, 1, 1, 1; %e A346871 ... %e A346871 Illustration of initial terms: %e A346871 Row 1: _ %e A346871 _| | %e A346871 |_ _| %e A346871 2 Semilength = 2 %e A346871 . %e A346871 Row 2: _ %e A346871 | | %e A346871 _ _|_| %e A346871 |_ _|1 Semilength = 3 %e A346871 2 %e A346871 . %e A346871 Row 3: _ %e A346871 | | %e A346871 | | %e A346871 _|_| %e A346871 _ _ _| Semilength = 5 %e A346871 |_ _ _|2 %e A346871 3 %e A346871 . %e A346871 Row 4: _ %e A346871 | | %e A346871 | | %e A346871 | | %e A346871 _|_| %e A346871 _| %e A346871 _ _ _ _| 1 Semilength = 7 %e A346871 |_ _ _ _|2 %e A346871 4 %e A346871 . %e A346871 Row 5: _ %e A346871 | | %e A346871 | | %e A346871 | | %e A346871 | | %e A346871 | | %e A346871 _ _|_| %e A346871 _| %e A346871 _|1 Semilength = 11 %e A346871 |1 %e A346871 _ _ _ _ _ _| %e A346871 |_ _ _ _ _ _|3 %e A346871 6 %e A346871 . %e A346871 The area (also the number of cells) of the successive diagrams gives A008864. %Y A346871 Row sums give A000040. %Y A346871 For the characteristic shape of sigma(A000079(n)) see A346872. %Y A346871 For the characteristic shape of sigma(A000217(n)) see A346873. %Y A346871 For the visualization of Mersenne numbers A000225 see A346874. %Y A346871 For the characteristic shape of sigma(A000384(n)) see A346875. %Y A346871 For the characteristic shape of sigma(A000396(n)) see A346876. %Y A346871 For the characteristic shape of sigma(A008588(n)) see A224613. %Y A346871 Cf. A000203, A008864, A018252, A237591, A237593, A245092, A249351, A262626. %K A346871 nonn,tabf %O A346871 1,1 %A A346871 _Omar E. Pol_, Aug 06 2021