[seqfan] Re: sigma(sigma(sigma(n)))/n < 3/2 ?
israel at math.ubc.ca
israel at math.ubc.ca
Sun Jun 16 22:32:48 CEST 2013
If n = p^k where p is prime, sigma(n) = 1 + p + ... + p^k ~ n. If that is a
prime power q^j (which is possible if k is even), sigma(sigma(n)) = 1 + q +
... + q^j ~ n. If 1 + q + ... + q^j is a prime power r^i (which again
requires j to be even), sigma(sigma(sigma(n))) = 1 + r + ... + r^i ~ n. So
we look for x which is an even power of a prime such that sigma(x) and
sigma(sigma(x)) are again even powers of primes. I looked at the primes p
<= 10^6 and powers j = 2,4,...,40. Of these the only case for which
sigma(p^j) was a square was 81.
Well, for k = 2 we'd have sigma(n) = 1 + p + p^2 < (1+p)^2 which is the
next square after n, so we need k >= 4.
If n = p^4, sigma(n) = 1 + p + p^2 + p^3 + p^4 and sqrt(sigma(n)) = p^2 +
p/2 + 3/3 + 5/(16 p) + ..., In fact p^2 + p/2 - 1/2 and p^2 + p/2 + 1/2 are
consecutive integers with (p^2 + p/2 - 1/2)^2 = p^4 + p^3 - (3/4) p^2 -
(1/2) p + 1/4 < sigma(p^4) and (p^2 + p/2 + 1/2)^2 = sigma(p^4) +
(1/4)(p+1)(p-3) > sigma(p^4) for p > 3. (Note that the case p=3 is our one
example 81 = 3^4). So we'll need k >= 6.
I've also ruled out k= 6, by a similar but somewhat more complicated
argument (I have to look separately at the cases p==1,3,5,7 mod 8).
I suspect there are no further examples, but I don't have a proof.
On Jun 16 2013, луиза уруджева wrote:
> 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 , Svetlana
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