



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
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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, HALFGOOD, 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,P1) 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=Ps.
When (x,y) is an existing partition and x is odd, the successor partition is (s,t), where t=y/2 and s=Pt.
Step 3. Repeat step 2 until you return to (1,P1).
He then classified P as either GOOD, HALFGOOD, or BAD as follows:
P is GOOD when every integer from 1 to P1 appears among the left parts of the set of generated partitions.
P is HALFGOOD when P does not meet the requirements for GOOD, but every integer from 1 to P1 appears somewhere in the set of generated partitions.
P is BAD when P does not meet the requirements for GOOD or HALFGOOD.
The sequence of GOOD odd prime integers is identical to A001122.
The sequence of HALFGOOD 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 HALFGOOD, 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



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



