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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, 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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A070090 A230642 A053473 * A322983 A208459 A144764
KEYWORD
nonn
AUTHOR
Howard Givner, Jun 17 2021
EXTENSIONS
Name edited by Felix Fröhlich, Jun 28 2021
STATUS
approved

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Last modified June 12 17:41 EDT 2024. Contains 373358 sequences. (Running on oeis4.)