[seqfan] Re: Add a Fibonacci number to get a prime

[Neil Sloane]

Well, I think we should certainly try to rescue Braxton's sequence. There
should be two sequences, one giving the actual Fibonacci numbers, as in his
original submission, and one giving their indices.  Using essentially his
definition, with -1 if no such Fibonacci number exists, I don't see any
problem.  We just stop at the 25th term, until such time as someone proves
that a(26) exists or does not exist.  Does that sound reasonable?
Best regards
Neil

Neil J. A. Sloane, Chairman, OEIS Foundation.
Also Visiting Scientist, Math. Dept., Rutgers University,
Email: njasloane at gmail.com



On Tue, Mar 28, 2023 at 9:11 AM Allan Wechsler <acwacw at gmail.com> wrote:

> In that case, modify the definition to be -1 in the case that n + F(k) is
> always composite. We still don't run into the philosophical difficulty
> encountered by Braxton, because we can use techniques like Robert
> Gerbicz's to *prove* the fact. (In fact this technique could rescue Braxton
> as well.)
>
> On Tue, Mar 28, 2023 at 6:40 AM Robert Gerbicz <robert.gerbicz at gmail.com>
> wrote:
>
> > "An informal probabilistic argument indicates that a(n) exists for all
> n."
> >
> > It is a false conjecture. Say for n=14475 for every k
> > 14475+fibonacci(k) is composite!
> >
> > Could be the smallest such n value, but really not checked.
> >
> > Set T=2*3*5*7*11*23*31=1647030, then the Fibonacci sequence is periodic
> mod
> > T using period=240.
> > So need to check only the first 240 Fibonacci numbers, and for
> > each of them T and 14475+fibonacci(k) is not coprime, and of course
> bigger
> > than 31, so 14475+fibonacci(k) is composite for the first 240 k values,
> but
> > then for every k.
> >
> > Pontus von Brömssen <pontus.von.bromssen at gmail.com> ezt írta (időpont:
> > 2023. márc. 28., K, 3:21):
> >
> > > I played around with the idea of the recently rejected sequence A361831
> > > (see version #29) and came up with the following variant. Let a(n) be
> the
> > > smallest k such that n+Fibonacci(k) is prime. The sequence starts (from
> > > n=0): 3, 1, 0, 0, 1, 0, 1, 0, 4, 3, 1. For example, the first Fibonacci
> > > number F such that 8+F is prime is F=3=Fibonacci(4), so a(8)=4.
> > >
> > > An informal probabilistic argument indicates that a(n) exists for all
> n.
> > An
> > > example where a(n) gets fairly large is a(6313)=25224. (The
> corresponding
> > > prime 6313+F(25224) has 5272 digits.)
> > >
> > > Considering that a closely related sequence was rejected, is this
> > sequence
> > > worth submitting?
> > >
> > > Best regards,
> > >
> > > Pontus
> > >
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> > >
> >
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