A345388 - OEIS

%I #21 May 15 2022 12:32:47

%S 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 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 0,2,1,1,0,0,1,0,2,2,2,0,0,2,2,2,0,2,2

%N a(n) = 0, 1, or 2 according to whether A065091(n), the n-th odd prime, is in A001122, A139035, or A268923, respectively.

%C 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 Description of the method:

%C 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 Procedure for categorizing an odd prime integer P:

%C Step 1. Begin with an initial partition (1,P-1) of P.

%C Step 2. Generate a successor partition, derived from an existing partition.

%C 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 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 Step 3. Repeat step 2 until you return to (1,P-1).

%C He then classified P as either GOOD, HALF-GOOD, or BAD as follows:

%C P is GOOD when every integer from 1 to P-1 appears among the left parts of the set of generated partitions.

%C 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 P is BAD when P does not meet the requirements for GOOD or HALF-GOOD.

%C The sequence of GOOD odd prime integers is identical to A001122.

%C The sequence of HALF-GOOD odd prime integers is identical to A139035.

%C The sequence of BAD odd prime integers is identical to A268923.

%H 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 For P=5, the generated partition set is:

%e   (1,4), (3,2), (4,1), (2,3), (1,4), and thus 5 is GOOD, so a(2)=0.

%e For P=7, the generated partition set is:

%e   (1,6), (4,3), (2,5), (1,6), and thus 7 is HALF-GOOD, so a(3)=1.

%e For P=17, the generated partition set is:

%e   (1,16), (9,8), (13,4), (15,2), (16,1), (8,9), (4,13), (2,15), (1,16),

%e   but 3, 5, 6, 7, 10, 11, 12, and 14 do not appear, and thus 17 is BAD, so a(6)=2.

%Y Cf. A001122, A065091, A139035, A268923.

%K nonn

%O 1,6

%A _Howard Givner_, Jun 17 2021

%E Name edited by _Felix Fröhlich_, Jun 28 2021