[math-fun] even Q(n)

[Christian G.Bower]

On 8/2/2005 Bill Gosper wrote to math-fun

> George Andrews kindly informs me that any fixed power of 2
> divides almost all Q(n) (usually written q(n), to confuse
> you with the nome).

This got me interested in tallying which power of 2 divides q(n)
(known in the EIS as A000009) so I sent in A114912:

A114912  2^a(n) divides A000009(n) but 2^(a(n)+1) does not.

 0, 0, 0, 1, 1, 0, 2, 0, 1, 3, 1, 2, 0, 1, 1, 0, 5, 1, 1, 1, 6, 2, 0, 3, 1, 1,
0, 6, 1, 8, 3, 2, 1, 6, 9, 0, 2, 3, 5, 1, 0, 2, 1, 1, 3, 11, 8, 1, 1, 6, 1, 0,
1, 10, 1, 1, 2, 0, 3, 6, 7, 2, 1, 9, 2, 3, 2, 1, 13, 1, 0, 5, 9, 1, 1, 1, 1,
0, 1, 3, 9, 2, 6, 1, 1, 6, 6, 1, 1, 1, 1, 11, 0, 5, 6, 1, 2, 8, 6, 1, 0, 1

The 0's of this sequences are the generalized pentagonal
numbers A001318.

The 1's had not been tallied yet, so I entered them as A114913.
Interesting thing is that the sequence is very similar to A111174

A114913  Values such that A114912(a(n))=1. Values such that A000009(a(n))==2
(mod 4).
 
 3, 4, 8, 10, 13, 14, 17, 18, 19, 24, 25, 28, 32, 39, 42, 43, 47, 48, 50, 52,
54, 55, 62, 67, 69, 73, 74, 75, 76, 78, 83, 84, 87, 88, 89, 90, 95, 99, 101,
103, 105, 108, 109, 112, 113, 118, 119, 123, 125, 127, 130, 132, 134, 138,
140, 143, 144, 147, 149, 153, 154, 157
 
 
A111174 Numbers n such that 24*n + 1 is prime.
 
 3, 4, 8, 10, 13, 14, 17, 18, 19, 24, 25, 28, 32, 39, 42, 43, 47, 48, 50, 52,
54, 55, 62, 67, 69, 73, 74, 75, 78, 83, 84, 87, 88, 89, 90, 95, 99, 103, 105,
108, 109, 112, 113, 118, 119, 123, 125, 127, 130, 132, 134, 138, 140, 143,
144, 147, 153, 154, 157, 158, 162 (list)  
 
 
matching in the first 28 terms.  A little more searching and it appears
that every number in A111174 is in A114913, but there are probably
infinitely many numbers in the latter not in the former.  (I admit I
didn't look that hard for an exception, only about 2000 or so.)
I wonder if there might be an interesting theorem in number theory.  I
looked at the Gordon/Ono paper

http://www.math.wisc.edu/~ono/reprints/018.pdf

and the Alladi paper

http://www.ams.org/tran/1997-349-12/
S0002-9947-97-01831-X/S0002-9947-97-01831-X.pdf

that rwg mentioned. Alladi mentions that all the members of A114913
are of the form of a (generalized) pentagonal number + a square, but
that's a much larger set, and is it even known that the 24n+1 primes
satisfy that requirement.

I looked a little at Dean Hickerson's 8/4/2005 posting about q(n) mod 64.
I didn't fully understand it, but it does not appear to have an obvious
connection to prime number theory.

The Alladi paper also mentions A001935 (calling it g(n)).

It speculates that g(n) has the same power of 2 property (while I believe
Gordon/Ono proves it.)  It also studies the 1 mod 2 and 2 mod 4 cases.
So I submitted the power of 2 sequence as A115247.  The 0's of that
sequence (the odd values of A001935) are the the triangle numbers
A000217.  I submitted the 1's (the A001935 2 mod 4 values) as A115248
and found a similar result as the A114913 case: this time with the
sequence A005123:

A005123  (( primes = 1 mod 8 ) - 1)/8.
 
 2, 5, 9, 11, 12, 14, 17, 24, 29, 30, 32, 35, 39, 42, 44, 50, 51, 54, 56, 57,
65, 71, 72, 74, 75, 77, 80, 84, 95, 96, 101, 107, 110, 116, 117, 119, 122,
126, 129, 131, 137, 141, 144, 149, 150, 152, 156, 161, 162, 165 (list)  
 
This time only the first 7 terms matched and then the next 8.  More
importantly, it appears that every term of A005123 is in A115248.
Alladi mentions that each term of A115248 is a triangle number +
a square.

P.S.

Anyone who studies sequences for a while realizes that partitions
have been studied a lot.  So it amazes me when I find a fairly basic
partition sequence is not yet in the EIS.

A001935 has 3 simple interpretations.

Partitions into non multiples of 4.
Partitions with no even part repeated.
Partitions with no part repeated more than 3 times.

Would you believe that none of the labeled versions of these are
yet in the EIS.  They will be soon though as I'm submitting them
as A115275-A115277.

Christian