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

Simon Colton simonco at dai.ed.ac.uk
Thu Aug 19 17:38:26 CEST 1999

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:


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

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.


Simon Colton.

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