tau(sigma(n))=2 -> tau(tau(n))=2

[Robert G. Wilson v]

Dear Simon,

        Using Mathematica and the formula:

Do[If[DivisorSigma[0, DivisorSigma[0, n]] ==
    DivisorSigma[0, DivisorSigma[1, n]], Print[n]], {n, 1, 50000}]

has the output of 1, 2, 4, 9, 16, 18, 25, 50, 64, 144, 289, 576, 578,
729, 1458, 1600, 1681, 2401, 2916, 3362, 3481, 3600, 4096, 4624, 4802,
5041, 6962, 7921, 9604, 10082, 10201, 11664, 15625, 15842, 17161, 18225,
18496, 20402, 21609, 24400, 26896, 27889, 28561, 29929, 31250, 34322,
36450, 36864, 43218.

Please notice that the two sides are equal to each other and not just
equal to 2.

The tau (tau (n)) = 2 for primes or for even powers of primes. Whereas
the tau (sigma (n)) = 2 for 2 and even powers of some primes.  More
specifically:
2, 4, 9, 16, 25, 64, 289, 729, 1681, 2401, 3481, 4096, 5041, 7921,
10201, 15625, 17161, 27889, 28561, 29929, 65536, 83521, 85849, 146689,
262144, 279841, 458329, 491401.

So unless I am reading your message incorrectly,  your supposition is
incorrect.

Sincerely yours,
Robert G. Wilson v

Simon Colton wrote:

> Dear Sequence Fans,
>
> On Tuesday, my computer program, HR, used the encyclopedia
> to spot the following, very nice, conjecture:
>
> For any integer, n, if the sum of divisors of n is prime,
> then the number of divisors of n will be prime.
>
> Or, more formally,
>
> tau(sigma(n))=2 -> tau(tau(n))=2.
>
> I've managed to prove this, and a proof is available here:
>
> http://www.dai.ed.ac.uk/~simonco/research/maths/HRConjecture.ps
>
> I would like to know whether any of you have seen this,
> (or a more general conjecture) before, and whether it has
> any implications or applications. Any comments greatly
> appreciated.
>
> For anyone interested, the conjecture was found by the program
> first inventing the concept of integers where the number
> of divisors is prime (soon to be added to the encyclopedia,
> I hope). I then asked it to inform me of any sequences
> from the encyclopedia which were subsequences of the new
> sequence, and literally the first one it returned was A023194,
> "sum of divisors of n is prime". Hence it had spotted that
> those integers where the sum of divisors is prime have a
> prime number of divisors. HR checked the conjecture up to
> 1,000,000 and I wrote a GAP program to continue the check
> up to 10^11. By the time the GAP program had finished, I'd
> proved the result.
>
> Cheers,
>
> Simon Colton.
> ---------------------------------
> http://www.dai.ed.ac.uk/~simonco/