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Numbers k such that 7*R_k + 2 is prime, where R_k = 11...1 is the repunit (A002275) of length k.
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%I #50 Aug 29 2024 23:33:54

%S 0,2,66,86,90,102,386,624,7784,18536,113757,135879

%N Numbers k such that 7*R_k + 2 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

%C Also numbers k such that (7*10^k + 11)/9 is prime.

%C Although perhaps a degenerate case, A002275 defines R(0)=0. Thus zero belongs in this sequence since 7*0 + 2 = 2 is prime. - _Robert Price_, Oct 28 2014

%C a(11) > 10^5. - _Robert Price_, Nov 22 2014

%C a(13) > 2*10^5. - _Tyler Busby_, Feb 01 2023

%H Makoto Kamada, <a href="https://stdkmd.net/nrr/7/77779.htm#prime">Prime numbers of the form 77...779</a>.

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

%F a(n) = A056693(n-1) + 1 for n>1.

%e If k = 2, we get (7*10^2 + 11)/9 = (700+11)/9 = 79, which is prime.

%t Do[ If[ PrimeQ[ 7(10^n - 1)/9 + 2], Print[n]], {n, 0, 5000}] (* _Robert G. Wilson v_, Oct 15 2004 *)

%t Do[ If[ PrimeQ[((7*10^n) + 11)/9], Print[n]], {n, 0, 8131}] (* _Robert G. Wilson v_, Sep 27 2004 *)

%t Select[Range[0, 700], PrimeQ[(7 10^# + 11) / 9] &] (* _Vincenzo Librandi_, Nov 22 2014 *)

%o (Magma) [n: n in [0..300] | IsPrime((7*10^n+11) div 9)]; // _Vincenzo Librandi_, Nov 22 2014

%Y Cf. A002275, A056693, A093404.

%K more,nonn,hard

%O 1,2

%A Julien Peter Benney (jpbenney(AT)ftml.net), Sep 14 2004

%E a(9) from Kamada link by _Ray Chandler_, Dec 23 2010

%E a(1)=0 added and Mathematica programs adapted by _Robert Price_, Oct 28 2014

%E a(11)-a(12) from _Tyler Busby_, Feb 01 2023