A345388 a(n) = 0, 1, or 2 according to whether A065091(n), the n-th odd prime, is in A001122, A139035, or A268923, respectively.

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, 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, 0, 2, 1, 1, 0, 0, 1, 0, 2, 2, 2, 0, 0, 2, 2, 2, 0, 2, 2

Offset

1,6

Comments

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.

Description of the method:

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.

Procedure for categorizing an odd prime integer P:

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

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

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.

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.

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

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

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

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.

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

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

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

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

Links

Table of n, a(n) for n=1..87.

James Tanton, How to fold things into thirds, sevenths, and thirty-sevenths!, video, May 14, 2021.

Example

For P=5, the generated partition set is:
  (1,4), (3,2), (4,1), (2,3), (1,4), and thus 5 is GOOD, so a(2)=0.
For P=7, the generated partition set is:
  (1,6), (4,3), (2,5), (1,6), and thus 7 is HALF-GOOD, so a(3)=1.
For P=17, the generated partition set is:
  (1,16), (9,8), (13,4), (15,2), (16,1), (8,9), (4,13), (2,15), (1,16),
  but 3, 5, 6, 7, 10, 11, 12, and 14 do not appear, and thus 17 is BAD, so a(6)=2.

Crossrefs

Cf. A001122, A065091, A139035, A268923.

Sequence in context: A230642 A053473 A380833 * A322983 A208459 A144764

Adjacent sequences: A345385 A345386 A345387 * A345389 A345390 A345391

Keyword

nonn

Author

Howard Givner, Jun 17 2021

Extensions

Name edited by Felix Fröhlich, Jun 28 2021

Status

approved