| Contribution from M. F. Hasler and F. Firoozbakht (mymontain(AT)yahoo.com), Oct 30 2009: (Start)
If Q is a perfect number such that gcd(Q, 3(3^a(n)-4))=1 then m=3^(a(n)-1)
(3^a(n)-4)Q is a solution of the equation sigma(x)=3(x+Q). This is a result of
the following theorem.
Theorem : If for a prime q, Q is a (q-1)-perfect number and p=q^k-q-1 is
a prime such that gcd(Q, p*q)=1, then m=p*q^(k-1)*Q is a solution of the
equation sigma(x)=q(x+Q). The proof is easy. (End)
Contribution from M. F. Hasler and F. Firoozbakht (mymontain(AT)yahoo.com), Dec 07 2009: (Start)
2 is the only even term of this sequence because if n is an even number
greater than 2 then 3^n-4=(3^(n/2)-2)*(3^(n/2)+2) is composite.
We have also found the following generalization of this theorem.
See comment lines of the sequence A171271.
Theorem : If for a prime q, Q is a (q-1)-perfect number and for some integers
k and m, p=q^k-m*q-1 is a prime such that gcd(Q, p*q)=1, then x=p*q^(k-1)*Q
is a solution of the equation sigma(x)=q(x+m*Q). The proof is easy. (End)
No further terms < 200000.
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