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

7, 29, 33550337, 137438691329

Offset: 1

Author

Labos Elemer, Jun 14 2001

Keywords

• more
• nonn

Comments

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

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.

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]

Links

• Mersenne Forum, Thread 10336
• C. Rivera, Puzzle 203

Crossrefs

Cf. A000396.

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

Programs

Mathematica

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 *)

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

Formula

P(p)*[P(p)+1]/2 + 1 is prime, where P(p) is a Mersenne prime. (corrected by Lekraj Beedassy, May 01 2009)