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

Tomasz Ordowski tomaszordowski at gmail.com
Tue Apr 14 20:23:57 CEST 2020

```Dear readers!

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.

```

More information about the SeqFan mailing list