[seqfan] sigma(sigma(sigma(n)))/n < 3/2 ?

луиза уруджева urudzheva-svetlana at mail.ru
Sun Jun 16 12:44:28 CEST 2013

sigma(sigma(sigma(n)))/n < 3/2   ?
Hello,   seqfans!
Let    n > 1.
It’s evident that q1(n)=sigma(n)/n can be arbitrarily close to 1 (if    n is large prime).
Similarly, q2(n)=sigma(sigma(n))/n can be arbitrarily close to 1. Sufficient (but not necessary) condition is  "n =p^2, where p and p^2+p+1 are primes".
And what about q3(n)=sigma(sigma(sigma(n)))/n ?    It can be close to 3/2 provided that p,    p^2+p+1    are primes    and p^2+p+2 = 2q, where q is prime.
My question is: can q3(n) be less than 3/2 ?
Necessary but not sufficient condition is "n, sigma(n) and sigma(sigma(n)) both are odd". Odd terms of A008848 give examples of such numbers.    
To find the numbers   n , for which sigma(sigma(sigma(n)))/n < 3/2, need to investigate the numbers of the type: the number n itself, sigma(n) and sigma(sigma(n)) are odd numbers without small divisors.
Please, could someone find relevant numbers.
Thanks ,

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