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

[Max Alekseyev]

Vladimir's reply:

---------- Forwarded message ----------
From: Letsko Vladimir <val-etc at yandex.ru>
Date: 2013/11/13
Subject: Fwd: [seqfan] Re: oddness of iterations of sigma()

Hello Charles!

Your arguments seem to me doubtful.

> To find an example of #2 one would first need to find an example of #1, of course.

Of course n satisfying example #2 need to be term of A231484. But...
Can you explain why sigma(sigma(sigma(n))) is need to be odd to
provide sigma(sigma(sigma(n)))/n < 3/2?


> 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.

Of course n cannot be divisible by 3.
However n is divisible by 3 does not implies 13 divides
sigma(sigma(n)). For instance sigma(3^6) is prime.


> 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.

Why?
Let n is divisible by 5^4, 31^4 and 167^4. Hence sigma(n) is divisible
by (11*71)^2. But sigma(11*71)^2 is not divisible by 3.
Of course sigma(31^4) and sigma(167^4) have other prime divisors, but
we may propose that they are balanced with sigma of other factors of
n.


> It's easy to show that 7 cannot divide the number;

Why?
For instance why n cannot be divisible by 7^4?


> 11 forces a factor of 7 which quickly makes it fail as well

Why?
Firstly divisor 7^4 (for instance) in sigma(n) does not provide
sigma(sigma(n)) is divisible by 3
Secondly sigma(11^6) (for instance) isn't divisible by 7.

> etc.

etc.
--
С уважением, Владимир Лецко


On Tue, Nov 12, 2013 at 10:48 AM, Charles Greathouse
<charles.greathouse at case.edu> wrote:
> 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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