A007505 Primes of form 3*2^n - 1.

2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, 26388279066623, 108086391056891903, 55340232221128654847, 226673591177742970257407, 59421121885698253195157962751, 30423614405477505635920876929023

Offset: 1

Author

N. J. A. Sloane, Robert G. Wilson v

Keywords

• nonn
• nice

Comments

a(1) = 2, define f(k) = 2k+1, then a(n+1) = least prime fff...(a(n)). After 383 the next terem is 6143. We have f(383) = 767 (composite), f(767) = 1535 (composite), f(1565)=3071(composite), f(3071) = 6143 (prime), hence the next term is 6143= ffff(383). - Amarnath Murthy, Jul 13 2005

If n is in the sequence and m=(n+1)/3 then m is a solution of the equation, sigma(x+sigma(x))=3x (*). Is it true that there is no other solution of (*)? - Farideh Firoozbakht, Dec 05 2005

References

• H. Riesel, Prime numbers and computer methods for factorization, Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, pp. 381-384.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..27
• Heiko Harborth, On h-perfect numbers, Annales Mathematicae et Informaticae, 41 (2013) pp. 57-62.
• Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
• Wilfrid Keller, List of primes k*2^n - 1 for k < 300
• Amelia Carolina Sparavigna, A recursive formula for Thabit numbers, Politecnico di Torino (Italy, 2019).
• Amelia Carolina Sparavigna, Composition Operations of Generalized Entropies Applied to the Study of Numbers, International Journal of Sciences (2019) Vol. 8, No. 4, 87-92.
• Eric Weisstein's World of Mathematics, Thabit ibn Kurrah Number
• Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

Crossrefs

Subsequence of A083329.

See A002235 for more terms.

Cf. A039687 (primes of the form 3*2^n+1).

Cf. A010051.

Programs

Haskell

a007505 n = a007505_list !! (n-1)
a007505_list = filter ((== 1) . a010051') a083329_list
-- Reinhard Zumkeller, Sep 10 2013

Magma

[a: n in [0..200] | IsPrime(a) where a is 3*2^n-1]; // Vincenzo Librandi, Mar 20 2013

Mathematica

Reap[For[n = 0, n <= 103, n++, If[PrimeQ[p = 3*2^n - 1], Sow[p]]]][[2, 1]] (* Jean-François Alcover, Dec 12 2012 *)
Select[Table[3 2^n - 1, {n, 0, 100}], PrimeQ] (* Vincenzo Librandi, Mar 20 2013 *)

PARI

for(n=0,100, if(isprime(t=3<Charles R Greathouse IV, Feb 07 2017

Formula

a(n) = 3*2^A002235(n)-1. - Zak Seidov, Jul 21 2016