A058959 Numbers k such that 3^k - 4 is prime.

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

Offset: 1

Author

Robert G. Wilson v, Jan 13 2001

Keywords

• nonn
• more

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 (q-1)-perfect number and p=q^k-q-1 is a prime such that gcd(Q, p*q)=1, then m=p*q^(k-1)*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^n-4=(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 (q-1)-perfect number and for some integers k and m, p=q^k-m*q-1 is a prime such that gcd(Q, p*q)=1, then x=p*q^(k-1)*Q is a solution of the equation sigma(x)=q(x+m*Q). The proof is easy. (End)

Links

• Farideh Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
• Henri & Renaud Lifchitz, PRP Records.

Crossrefs

Cf. A014224, A171271. - M. F. Hasler and Farideh Firoozbakht, Dec 07 2009

Programs

Magma

[n: n in [1..10^3]| IsPrime(3^n-4)]; // Vincenzo Librandi, Mar 24 2015

Mathematica

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

PARI

for(n=1,10^3,if(ispseudoprime(3^n-4),print1(n,", "))) \\ Derek Orr, Mar 06 2015

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