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August 8th, 2018, 03:10 AM   #1
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Math Focus: Prime numbers
Primes of the form Sigma(Sigma(x))

Hi,
I checked for the primes of the form sigma(sigma(x)). (Sigma(x) : sum of divisors of x) and it is prime for the x values x>1;
x=3, 2667, 3937,57337,172011,253921 and all the primes of this form I found are Mersenne primes.
sigma(sigma(3))=7
......... ............(2667)=8191 :2^13-1
(3937)=8191
(57337)=131071 :2^17-1
(172011)=524287:2^19-1
(253921)=5242872^19-1

and all the x values>3 are products of Mersenne primes; 2667=3*7*127,
3937=31*127
,...
253921=31*8191
I know that if Sigma(x) is prime than x has to be a square. So if Sigma(Sigma(x)) is prime than Sigma(x) has to be a square.
I couldn't find any prime of this form that is not a Mersenne prime. Are all the primes of this form Mersenne primes? I don't know whether there is a proof about it; can you find a counter-example for it and what do you think?

Last edited by skipjack; August 8th, 2018 at 08:51 AM.
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