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Numbers k such that 2*R_k - 1 is prime, where R_k = 11...1 is the repunit (A002275) of length k.
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%I #49 Apr 29 2024 09:30:30

%S 4,18,100,121,244,546,631,1494,2566,8088,262603,282948,359860

%N Numbers k such that 2*R_k - 1 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

%C Also numbers k such that (2*10^k-11)/9 is prime.

%C Larger values correspond to strong pseudoprimes.

%C a(11) > 10^5. - _Robert Price_, Sep 06 2014

%H Makoto Kamada, <a href="https://stdkmd.net/nrr/2/22221.htm#prime">Prime numbers of the form 22...221</a>.

%H <a href="/index/Pri#Pri_rep">Index entries for primes involving repunits</a>

%F a(n) = A056660(n) + 1.

%e a(1) = 4 because 2*(10^4-1)/9-1 = 2221 is prime.

%e a(2) = 18 means that 222222222222222221 is prime.

%p select(t -> isprime(2*(10^t-1)/9-1),[$1..1000]); # _Robert Israel_, Sep 07 2014

%t Do[ If[ PrimeQ[2(10^n - 1)/9 - 1], Print[n]], {n, 0, 7000}] (* _Robert G. Wilson v_, Oct 14 2004; fixed by _Derek Orr_, Sep 06 2014 *)

%o (PARI) for(n=1, 10^4, if(ispseudoprime(2*(10^n-1)/9-1), print1(n,", "))) \\ _Derek Orr_, Sep 06 2014

%o (Python)

%o from sympy import isprime

%o def afind(limit):

%o n, twoRn = 1, 2

%o for n in range(1, limit+1):

%o if isprime(twoRn-1): print(n, end=", ")

%o twoRn = 10*twoRn + 2

%o afind(700) # _Michael S. Branicky_, Apr 18 2021

%Y Cf. A084831, A096503-A096508, A096841-A096846, A002275, A056660.

%K more,nonn

%O 1,1

%A _Jason Earls_, Jun 05 2003

%E a(8) from _Labos Elemer_, Jul 15 2004

%E a(10) from Kamada data by _Robert Price_, Sep 06 2014

%E a(11)-a(13) from Kamada data by _Tyler Busby_, Apr 29 2024