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Numbers k such that 7*R_(k+2) - 5*10^k is prime, where R_k = 11...1 is the repunit (A002275) of length k.
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%I #26 Apr 17 2024 15:20:54

%S 1,4,10,28,49,64,79,109,169,1270,5638,6812,7951,11737,16360,22840,

%T 25394,33394

%N Numbers k such that 7*R_(k+2) - 5*10^k is prime, where R_k = 11...1 is the repunit (A002275) of length k.

%C Also, numbers k such that (655*10^k - 7)/9 is prime.

%C Terms from Kamada.

%C a(18) > 30000.

%H Makoto Kamada, <a href="https://stdkmd.net/nrr/abaaa.htm">Near-repdigit numbers of the form ABAA...AA</a>.

%H Makoto Kamada, <a href="https://stdkmd.net/nrr/7/72777.htm#prime">Prime numbers of the form 7277...77</a>.

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

%e For k=4, 7*R_6 - 5*10^2 = 777777 - 50000 = 727777 which is prime.

%t Select[Range[0, 30000], PrimeQ[(655*10^#-7)/9 ] &]

%o (PARI) for(n=0,200,if(isprime((655*10^n-7)/9),print1(n,", "))) \\ _Derek Orr_, Apr 14 2015

%o (Magma) [n: n in [0..400] | IsPrime((655*10^n-7) div 9)]; // _Vincenzo Librandi_, Apr 15 2015

%Y Cf. A002275.

%K more,hard,nonn

%O 1,2

%A _Robert Price_, Apr 14 2015

%E a(18) from Kamada data by _Tyler Busby_, Apr 17 2024