[seqfan] Re: Proof For A269254

Sat Oct 28 02:19:13 CEST 2017

After: http://list.seqfan.eu/pipermail/seqfan/2017-October/018030.html
And: http://list.seqfan.eu/pipermail/seqfan/2017-October/018052.html

Time's up! In less than 15 seconds, the following algorithm completes a
brute force proof for all Chebyshev cases less than cutoff N=10,000:

http://community.wolfram.com/groups/-/m/t/1209741?p_p_auth=9I4iR9TT

and the output table is here:

https://ptpb.pw/txWp

This extends the data set to include induction zero-sums for 2< n < N,
which are printed in the last column of the table above. These sums seem to
also have some interesting structure, apparently as palindromes.

One important observation from the tabulated data is that this algorithm
does produce false positives, if we are searching for values a(n)=-1. These
occur in rows where the first column reads "False". The sequence s_k can
very well visit a prime in the first few values, yet admit a factor
decomposition for all higher values. With this data it seems more natural
not to specially regard values a(n)=-1. Rather we should simply look for
cases where the recursion relations admit factorization. If the initial
conditions have a prime, so what?

In addition to verifying the induction, we also need to check the strictly
increasing property. This task is relatively easy. The second truth value
indicates whether or not the initial conditions strictly increase. When the
increasing check passes, it's easy to see that the factor sequences never
visit a 1.

The third truth value indicates that each of the m-subsequences satisfy the
same recursion relations.

The remaining table columns are: signatures for the recursion relations
followed by one zero for successful evaluation of the base case.

Thanks,