[seqfan] oddness of iterations of sigma()

[Max Alekseyev]

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