

A061644


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


4




OFFSET

1,1


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^(p1)*(2^p1), 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)) = 2x2. Proof: take t=2 in theorem related to the sequence A093034. [M. F. Hasler and Farideh Firoozbakht, Sep 09 2014]


LINKS

Table of n, a(n) for n=1..4.
Mersenne Forum, Thread 10336
C. Rivera, Puzzle 203


FORMULA

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


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 *)


PROG

(PARI) forprime(p=2, 100, P=2^p1; Q=P*(P+1)/2+1; if(isprime(P)&&isprime(Q), print1(Q, ", "))) \\ Edward Jiang, Sep 10 2014


CROSSREFS

Cf. A000396.
Analogous right and left multiple perfect numbers are in A093034, A094467.
Sequence in context: A122119 A300528 A157422 * A053621 A339098 A210107
Adjacent sequences: A061641 A061642 A061643 * A061645 A061646 A061647


KEYWORD

more,nonn


AUTHOR

Labos Elemer, Jun 14 2001


STATUS

approved



