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 A345388 #21 May 15 2022 12:32:47 %S A345388 0,0,1,0,0,2,0,1,0,2,0,2,2,1,0,0,0,0,1,2,1,0,2,2,0,1,0,2,2,2,0,2,0,0, %T A345388 2,2,0,1,0,0,0,1,2,0,1,0,2,0,2,2,1,2,2,2,1,0,1,2,2,2,0,2,1,2,0,2,2,0, %U A345388 0,2,1,1,0,0,1,0,2,2,2,0,0,2,2,2,0,2,2 %N A345388 a(n) = 0, 1, or 2 according to whether A065091(n), the n-th odd prime, is in A001122, A139035, or A268923, respectively. %C A345388 The three OEIS sequences A001122, A139035, and A268923 are implicitly described in a Zoom lecture that was given May 14, 2021, by James Tanton. Here is a link to the video, followed by a description of how the sequences can be obtained by carrying out the procedure that the speaker described in his talk. %C A345388 Description of the method: %C A345388 James Tanton defined GOOD, HALF-GOOD, and BAD odd prime integers and a procedure for determining which of the three categories an odd prime integer belongs to. %C A345388 Procedure for categorizing an odd prime integer P: %C A345388 Step 1. Begin with an initial partition (1,P-1) of P. %C A345388 Step 2. Generate a successor partition, derived from an existing partition. %C A345388 When (x,y) is an existing partition and x is even, the successor partition is (s,t), where s=x/2 and t=P-s. %C A345388 When (x,y) is an existing partition and x is odd, the successor partition is (s,t), where t=y/2 and s=P-t. %C A345388 Step 3. Repeat step 2 until you return to (1,P-1). %C A345388 He then classified P as either GOOD, HALF-GOOD, or BAD as follows: %C A345388 P is GOOD when every integer from 1 to P-1 appears among the left parts of the set of generated partitions. %C A345388 P is HALF-GOOD when P does not meet the requirements for GOOD, but every integer from 1 to P-1 appears somewhere in the set of generated partitions. %C A345388 P is BAD when P does not meet the requirements for GOOD or HALF-GOOD. %C A345388 The sequence of GOOD odd prime integers is identical to A001122. %C A345388 The sequence of HALF-GOOD odd prime integers is identical to A139035. %C A345388 The sequence of BAD odd prime integers is identical to A268923. %H A345388 James Tanton, <a href="https://mathtube.org/lecture/video/how-fold-things-thirds-sevenths-and-thirty-sevenths">How to fold things into thirds, sevenths, and thirty-sevenths!</a>, video, May 14, 2021. %e A345388 For P=5, the generated partition set is: %e A345388 (1,4), (3,2), (4,1), (2,3), (1,4), and thus 5 is GOOD, so a(2)=0. %e A345388 For P=7, the generated partition set is: %e A345388 (1,6), (4,3), (2,5), (1,6), and thus 7 is HALF-GOOD, so a(3)=1. %e A345388 For P=17, the generated partition set is: %e A345388 (1,16), (9,8), (13,4), (15,2), (16,1), (8,9), (4,13), (2,15), (1,16), %e A345388 but 3, 5, 6, 7, 10, 11, 12, and 14 do not appear, and thus 17 is BAD, so a(6)=2. %Y A345388 Cf. A001122, A065091, A139035, A268923. %K A345388 nonn %O A345388 1,6 %A A345388 _Howard Givner_, Jun 17 2021 %E A345388 Name edited by _Felix Fröhlich_, Jun 28 2021