

A058959


Numbers n such that 3^n  4 is prime.


26



2, 3, 5, 21, 31, 37, 41, 53, 73, 101, 175, 203, 225, 455, 557, 651, 1333, 4823, 20367, 32555, 52057, 79371, 267267, 312155
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OFFSET

1,1


COMMENTS

From M. F. Hasler and Farideh Firoozbakht, Oct 30 2009: (Start)
If Q is a perfect number such that gcd(Q, 3(3^a(n)4))=1 then m=3^(a(n)1) (3^a(n)4)Q is a solution of the equation sigma(x)=3(x+Q). This is a result of the following theorem.
Theorem : If for a prime q, Q is a (q1)perfect number and p=q^kq1 is a prime such that gcd(Q, p*q)=1, then m=p*q^(k1)*Q is a solution of the equation sigma(x)=q(x+Q). The proof is easy. (End)
From M. F. Hasler and Farideh Firoozbakht, Dec 07 2009: (Start)
2 is the only even term of this sequence because if n is an even number greater than 2 then 3^n4=(3^(n/2)2)*(3^(n/2)+2) is composite.
We have also found the following generalization of this theorem. See comment lines of the sequence A171271.
Theorem : If for a prime q, Q is a (q1)perfect number and for some integers k and m, p=q^km*q1 is a prime such that gcd(Q, p*q)=1, then x=p*q^(k1)*Q is a solution of the equation sigma(x)=q(x+m*Q). The proof is easy. (End)


LINKS

Table of n, a(n) for n=1..24.
F. Firoozbakht, M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1
Henri & Renaud Lifchitz, PRP Records.


MATHEMATICA

Do[ If[ PrimeQ[3^n  4], Print[n] ], {n, 1, 3000} ]


PROG

(PARI) for(n=1, 10^3, if(ispseudoprime(3^n4), print1(n, ", "))) \\ Derek Orr, Mar 06 2015
(MAGMA) [n: n in [1..10^3] IsPrime(3^n4)]; // Vincenzo Librandi, Mar 24 2015


CROSSREFS

Cf. A014224, A171271.  M. F. Hasler and Farideh Firoozbakht, Dec 07 2009
Sequence in context: A291673 A076383 A024766 * A065398 A084838 A051694
Adjacent sequences: A058956 A058957 A058958 * A058960 A058961 A058962


KEYWORD

nonn,more


AUTHOR

Robert G. Wilson v, Jan 13 2001


EXTENSIONS

a(18)=4823, corresponding to a certified prime, from Ryan Propper, Jun 30 2005
a(19)=20367 from Ray Chandler, Jul 25 2011
a(20)=32555, a(21)=52057 from Henri Lifchitz, Jan 2005
a(22)=79371 from Ray Chandler, Jul 25 2011
a(23)=267267 from Roman Ilyukhin, Oct 17 2014
a(24)=312155 from Roman Ilyukhin, Feb 28 2015


STATUS

approved



