cp's OEIS Frontend

A061644 "Right perfect numbers": primes of the form 1 + a perfect number.

%I A061644 #26 Sep 11 2014 04:33:38
%S A061644 7,29,33550337,137438691329
%N A061644 "Right perfect numbers": primes of the form 1 + a perfect number.
%C A061644 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 A061644 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 A061644 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 A061644 Mersenne Forum, <a href="http://www.mersenneforum.org/showthread.php?t=10336">Thread 10336</a>
%H A061644 C. Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_203.htm">Puzzle 203</a>
%F A061644 P(p)*[P(p)+1]/2 + 1 is prime, where P(p) is a Mersenne prime. (corrected by _Lekraj Beedassy_, May 01 2009)
%t A061644 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 A061644 (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 A061644 Cf. A000396.
%Y A061644 Analogous right and left multiple perfect numbers are in A093034, A094467.
%K A061644 more,nonn
%O A061644 1,1
%A A061644 _Labos Elemer_, Jun 14 2001