# [seqfan] Re: Unexpected properties of A003524 (divisors of 2^{13-1} - 1)

Tomasz Ordowski tomaszordowski at gmail.com
Wed Apr 15 10:51:57 CEST 2020

```P.S. See my draft
http://oeis.org/draft/A334133

T. Ordowski
____________
https://oeis.org/history/view?seq=A334133&v=26

wt., 14 kwi 2020 o 20:23 Tomasz Ordowski <tomaszordowski at gmail.com>
napisał(a):

>
> Let a(n) = gpf(A111076(n)^lambda(n) - 1), for n > 2,
> where gpf(n) = A006530(n) is the greatest prime factor of n
> and lambda(n) = A002322(n) is the Carmichael function of n.
> See above all https://oeis.org/A111076
>
> Let's define: Numbers n > 2 such that a(n) = gpf(lambda(n) + 1).
> 3, 5, 6, 9, 10, 12, 13, 15, 16, 20, 21, 24, 30, 35, 39, 40, 45, 60, 63,
> 65,
> 80, 91, 105, 117, 120, 195, 240, 273, 315, 455, 585, 819, 1365, 4095.
> Probably complete.
>
> (*) Conjecture: The above odd numbers n are
> 3, 5, 9, 13, 15, 21, 35, 39, 45, 63, 65, 91, 105,
> 117, 195, 273, 315, 455, 585, 819, 1365, 4095
> odd numbers k such that gpf(2^m - 1) = gpf(m+1),
> where m = ord_{k}(2) = A002326((k-1)/2)
> is the multiplicative order of 2 mod 2k+1.
>
> Cf. A003524 except 1 and 7.
> See https://oeis.org/A003524 (all divisors of 2^12-1).
> Are these all such odd numbers? If so, how to prove this fact.
> Note that p = 13 is the greatest prime p for which gpf(2^{p-1}-1) = p.
>
> Greetings to everyone affected by the coronavirus pandemic!
>
> Best regards,
>
> Thomas Ordowski
> _______________
> (*) This conjecture was put forward by Amiram Eldar
> in a private message to me.
>

```