[seqfan] Fwd: Sequences taken from straight lines through the prime spiral.

[David Sycamore]

Message slightly edited


Begin forwarded message:

> From: David Sycamore <djsycamore at yahoo.co.uk>
> Date: 24 July 2018 at 14:13:07 GMT+2
> To: seqfan at list.seqfan.eu
> Subject: Sequences taken from straight lines through the prime spiral.
> 
> 
> Consider the primes arranged in a clockwise square spiral as shown below, centred on prime(1)=2 (Ulam type spiral, but  primes only).
> 
> There are already several sequences in oeis based on this spiral (A053999, A0854568, A054553...). These all go outwards from the same central starting point (2) in so called
> « spokes ».
> 
> The following sequences are based not on spokes but on the main (compass) axes (diagonals plus horizontal and vertical lines through the central first prime 2), arranged  in order of size. 
> 
> The numbers in the NW~SE diagonal are: 2,5,17,41,73,129,191.....
> Looking this up in the data base we find A122566, a(n)=prime(k^2+k+1), but no mention in Name of any connection to the prime spiral. The indices of this sequence of primes seems to be A0002061 (central polygonal numbers a(n)=n^2-n+1), n>=1, which, judging by Tedeschi’s comment is related to spiral (sorted) numbers. 
> 
> The other (SW~NE) diagonal is:
> 2,11,23,59,97,157,227.......
> 
> This is not in the data base. However  the associated sequence of indices (1,5,9,17,25,37,49....) is A080335 
> « Diagonal in maze arrangement of natural numbers ». The name sounds like there could be some connection with this idea but it’s not certain. Can anyone explain this sequence?
> 
> The (SW~NE) diagonal primes could be given as prime(A080335(n)), or
> prime(3+4*k+2*k^2 -(-1)^n)/2). 
> 
> If we do the same for the central vertical (N~S) line we get  2,7,19,47,83,139,199,283,389.. which is not in the data base, although the associated sequence of prime indices (A267682) seems to be there, (Rule 201 etc). Its not clear (to me at least) what A267682 is all about and if it could have any connection with the prime spiral. Can anyone throw some light on this? 
> 
> Doing the same for the central horizontal (E~W) line gives 2,3,13,31,67,107,173...
> with prime indices 1,2,6,11,19,28,40...
> Neither of these sequences appear in the database.
> 
> In summary, of the 4 main lines looked at here, only one is already in the data base (though not mentioned as such in Name), two others could perhaps be related to  existing sequences of indices, and one is completely absent (terms and indices). 
> 
> Is this topic of interest, and should these sequences be in the oeis?
> 
>    73 79  83  89  97
>    71  17  19  23  29
>    67  13    2   3   31
>    61  11     7   5  37
>    59 53   47  43 41
> 
> The above four sequences are all centred on 2, though obviously many other straight line sequences of size ordered terms are possible. 
> 
> I would be interested to hear any comment on this subject. 
> Thanks
> David.
>