A061644 - OEIS

%I #26 Sep 11 2014 04:33:38

%S 7,29,33550337,137438691329

%N "Right perfect numbers": primes of the form 1 + a perfect number.

%C Readers of Rivera's web page (which I believe was indirectly based on this entry) later showed that there are no more cases among the first 39 perfect numbers. - _N. J. A. Sloane_, May 25 2004. The latest news is that there are no more cases among the first 44 perfect numbers. - _M. F. Hasler_, Jun 05 2008

%C So of the 44 known perfect numbers P=2^(p-1)*(2^p-1), P+1 is only prime for p=2,3,13 and 19.

%C If p is in the sequence then for each positive integer k, p^k is a solution to the equation sigma(phi(x)) = 2x-2. Proof: take t=2 in theorem related to the sequence A093034. [_M. F. Hasler_ and _Farideh Firoozbakht_, Sep 09 2014]

%H Mersenne Forum, <a href="http://www.mersenneforum.org/showthread.php?t=10336">Thread 10336</a>

%H C. Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_203.htm">Puzzle 203</a>

%F P(p)*[P(p)+1]/2 + 1 is prime, where P(p) is a Mersenne prime. (corrected by _Lekraj Beedassy_, May 01 2009)

%t pn={6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216};lst={};Do[p=pn[[n]]+1;If[PrimeQ[p],AppendTo[lst,p]],{n,Length[pn]}];lst... and/or...PerfectNum[n_]:=Plus@@Divisors[n]/2;lst={};Do[p=PerfectNum[n];If[p==n&&PrimeQ[p+1],AppendTo[lst,p+1]],{n,10!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Jan 27 2009 *)

%o (PARI) forprime(p=2,100,P=2^p-1;Q=P*(P+1)/2+1;if(isprime(P)&&isprime(Q),print1(Q,","))) \\ _Edward Jiang_, Sep 10 2014

%Y Cf. A000396.

%Y Analogous right and left multiple perfect numbers are in A093034, A094467.

%K more,nonn

%O 1,1

%A _Labos Elemer_, Jun 14 2001