OFFSET
1,1
COMMENTS
This sequence is infinite, assuming Dickson's conjecture. In fact, the conjecture implies that there are infinitely many terms of this sequence divisible by any fixed prime p. - Charles R Greathouse IV, May 08 2011
Related to the Rassias Conjecture ("for any odd prime p there are primes q < r such that p*q = q+r+1") setting n = q*r, a = q+r+1. The sequence includes the cases with p = q (or p = r). Generalization can be achieved by removing the semiprimality condition or accepting gcd(n,a)=1. The differential equation in its general form n' = a + 1 includes Primary Pseudoperfect numbers, i.e., n' = n-1 (A054377).
LINKS
For Rassias conjecture: Preda Mihăilescu, Review of Problem Solving and Selected Topics in Number Theory, Newsletter of the European Mathematical Society, March 2011, p. 46.
FORMULA
Semiprimes pq with (p+q+1)/p prime. - Charles R Greathouse IV, May 08 2011
EXAMPLE
For n=6, 6' = 5, a = 5-1 = 4, gcd(4,6)=2, so 6 is a term.
MAPLE
der:=n->n*add(op(2, p)/op(1, p), p=ifactors(n)[2]);
# for quick reference only
seq(`if`(bigomega(i)=2 and bigomega(der(i)+1)=2 and gcd(i, der(i)+1)>1, i, NULL), i=1..2000);
PROG
(PARI) find(lim)=my(v=List()); forprime(p=2, sqrtint(lim\2), forstep(q=2*p-1, lim\p, p+p, if(isprime(q\p+2)&isprime(q), listput(v, p*q)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, May 08 2011
CROSSREFS
KEYWORD
nonn
AUTHOR
Giorgio Balzarotti, May 07 2011
STATUS
approved