[seqfan] Re: oddness of iterations of sigma()

[Charles Greathouse]

To find an example of #2 one would first need to find an example of #1, of
course. Additionally it cannot be divisible by 3, since then 13 divides
sigma(n) and so sigma(13^2) divides sigma(sigma(n)) which forces the ratio
to be > 1.56. A similar but more complex argument shows that is cannot be
divisible by 5 either, because 5 | n ensures sigma(sigma(n)) is divisible
by 3 which together with the other factors is too much. It's easy to show
that 7 cannot divide the number; 11 forces a factor of 7 which quickly
makes it fail as well; etc. The smallest primes I cannot exclude in this
manner are

41, 71, 101, 251, 383, 479, 509, 587, 701, 761, 773, 797, 827, 839, 929,
1091, 1097, 1163, 1193, 1217, 1289, 1373, 1487, 1499, 1553, 1559, 1583,
1709, 1811, 1889, 1931, 2129, 2309, 2351, 2411, 2693, 2729, 2789, 2957,
2969, 3011, 3041, 3191, 3209, 3221, 3449, 3491, 3557, 3671, 3863, 3881,
4019, 4157, 4217, 4259, 4409, 4679, 4721, 4751, 4817, 4877, 4973, 5039,
5081, 5087, 5351, 5507, 5717, 5867, 5981, 6047, 6389, 6473, 6551, 6569,
6599, 6653, 6791, 6833, 6959, 7253, 7433, 7547, 7841, 7853, 7883, 7937,
8093, 8237, 8387, 8501, 8543, 8627, 8681, 8741, 8753, 8807, 8963, 9323,
9533, 9539, 9689, 9719, 9743, ...

This should narrow the sample space considerably.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University


On Tue, Nov 12, 2013 at 10:09 AM, Harvey P. Dale <hpd at hpdale.org> wrote:

>         In response to the first question, there are no such odd numbers
> up to 50 million.
>         Best,
>         Harvey
>
> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Max
> Alekseyev
> Sent: Tuesday, November 12, 2013 8:59 AM
> To: seqfaneu
> Subject: [seqfan] oddness of iterations of sigma()
>
> Vladimir Letsko is unfortunate in his attempts to join SeqFan maillist.
> So he asked me to forward his question; please 'carbon copy' your replies
> to his email.
>
> ---------- Forwarded message ----------
> From: Letsko Vladimir <val-etc at yandex.ru>
> Date: 2013/11/12
>
>
> Hello SeqFans!
>
> Does anybody know the answer on some questions associated with A231484?
>
> 1. Does there exists an odd number n > 1 for which sigma(n),
> sigma(sigma(n)) and sigma(sigma(sigma(n))) are odd too?
>
> 2. Does there exists a number n > 1 such that sigma(sigma(sigma(n)))/n <
> 1.5?
>
> 3. Note that sigma(3^4) = 11^2. Does there exists another pair (p,r) such
> that p is prime, r > 1 and sigma(p^r) = q^s where q is prime and s > 1?
>
> Best regards,
> Vladimir Letsko
>
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