[seqfan] Re: Nice conjecture from Enrique Navarrete re divisors of triangular numbers A111273

[Neil Sloane]

Peter,  Yes, very nice!  This proves Enrique's conjecture in A111273, and
my conjecture in A309195.
Can you you go ahead and edit these entries?  Congratulations!

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com



On Fri, Jul 26, 2019 at 1:54 AM Peter Munn <techsubs at pearceneptune.co.uk>
wrote:

> Does this attempt at a proof stand up?:
>
> For odd k, k appears by A111273(k). Proof: choose m such that k-1 <= m <=
> k and T_m is odd. k is a divisor of T_m and (by induction) all smaller odd
> divisors have occurred earlier, so a(m) = k if k has not occurred earlier.
>
> For even k, k appears by A111273(2k-1), as k divides T_(2k-1) and by
> induction all smaller divisors have occurred earlier.
>
> So, for odd prime p, the first triangular number it divides is T_(p-1) =
> p*(p-1)/2. But (p-1)/2 and any smaller divisors have occurred by (p-1)-1,
> so A111273(p-1) = p.
>
> In respect of the conjecture in A309196 that after A111273(1), the
> smallest missing value is always even, note that even numbers can only
> appear in A111273 at indices congruent to 0 and 3 modulo 4. So as
> A111273(4) is not even, for n >= 4 there is always an even number less
> than or equal to n that has not appeared by A111273(n), whereas all such
> odd numbers have.
>
> Best Regards,
>
> Peter
>
> On Wed, July 24, 2019 8:40 pm, israel at math.ubc.ca wrote:
> > No, q is not a(q-1) in general, just for odd primes. For example,
> > if p = 127, the divisors of t_126 less than p are 1 = a(1), 3 = a(2),
> > 7 = a(6), 9 = a(9), 21 = a(14), 63 = a(62).
> > Cheers,
> > Robert
>
> > On Jul 24 2019, Fred Lunnon wrote:
> >>  By induction: if q|(p-1) then q = a(q-1) has appeared earlier;
> >> the only
> >> divisor of t_p remaining is p itself, so a(p-1) = p . QED WFL
>
> >>On 7/24/19, Neil Sloane <njasloane at gmail.com> wrote:
> >>> E.N. noticed that in A111273 (definition: a(n) = smallest divisor of
> t_n = n(n+1)/2 that has not yet appeared in the sequence), it seems
> that
> >>> a(p-1)=p for all odd primes p. Surely this can't be a hard problem? It
> would follow if we knew that the smallest missing number (smn) in
> A111273 (which is now A309195) is always >n/2.
>
>
>
>
>
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