1,1

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 (njas(AT)research.att.com), May 25 2004. The latest news is that there are no more cases among the first 44 perfect numbers - Maximilian Hasler, Jun 05 2008.

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. Rivera, Puzzle 203

Mersenne Forum, Thread 10336

P(p)*[P(p)+1]/2 is prime, where P(p) is a Mersenne prime.

P(p)*[P(p)+1]/2 + 1 is prime, where P(p) is a Mersenne prime.(Rectified) [From Lekraj Beedassy (blekraj(AT)yahoo.com), May 01 2009]

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 [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jan 27 2009]

Cf. A000396.

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

Sequence in context: A135629 A122119 A157422 this_sequence A053621 A018831 A063128

Adjacent sequences: A061641 A061642 A061643 this_sequence A061645 A061646 A061647

more,nonn

Labos E. (labos(AT)ana.sote.hu), Jun 14 2001