[seqfan] Re: square loop generalised to power loop

[Neil Fernandez]

In message <201910121813.x9CIDQP13739 at crypt.org>, hv at crypt.org writes

>After a recent puzzle in New Scientist.
>
>The integers 1 .. 32 can be arranged in a loop such that each consecutive
>pair sums to a square:
>  32 4 21 28 8 1 15 19 26 23 2 14 22 27 9 16
>  20 29 7 18 31 5 11 25 24 12 13 3 6 30 19 17
>
>My trial code to test for this finds n = 32 is the smallest for which this
>is possible, and finds solutions for each of 32 to 44; however the code
>is becoming unusably slow as n increases.
>
>My suspicion is that it is possible precisely for n >= 32, can someone
>prove this, or at least show an upper bound for an n for which the loop
>is not possible?
>
>If we require only a sequence rather than a loop, the first solution
>occurs with n = 15:
>  8 1 15 10 6 3 13 12 4 5 11 14 2 7 9 
>.. and it appears there are solutions for n in { 15, 16, 17, 23 } and
>all n >= 25 (tested up to n = 47).
>
>I would guess that the two examples might be of interest in the OEIS, but
>the sets of values of n for which loops or sequences are (or are not)
>possible would not be suitable as OEIS sequences.

Hi Hugo,

Another sequence is

a(n):= smallest k such that {1, 2 ,..., k} can be arranged in a loop in
which every consecutive pair sums to an nth power

The sequence begins 1, 32, ...

Neil