A007505 Primes of form 3*2^n - 1. (Formerly M1395)

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

Offset

1,1

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

Formula

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

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 *)

Prog

(Magma) [a: n in [0..200] | IsPrime(a) where a is 3*2^n-1]; // Vincenzo Librandi, Mar 20 2013
(Haskell)
a007505 n = a007505_list !! (n-1)
a007505_list = filter ((== 1) . a010051') a083329_list
-- Reinhard Zumkeller, Sep 10 2013
(PARI) for(n=0, 100, if(isprime(t=3<<n-1), print1(t", "))) \\ Charles R Greathouse IV, Feb 07 2017

Crossrefs

See A002235 for more terms.

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

Cf. A010051, subsequence of A083329.

Sequence in context: A084403 A261201 A055011 * A246492 A059411 A126017

Adjacent sequences: A007502 A007503 A007504 * A007506 A007507 A007508

Keyword

nonn,nice

Author

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

Status

approved