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Numbers k such that R(k) + 2*10^floor(k/2-1) is prime, where R(k) = (10^k-1)/9 (repunit: A002275).
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%I #20 Feb 03 2023 14:36:11

%S 2,3,5,8,9,39,78,81,155,249,387,395,510,711,1173,1751,10245

%N Numbers k such that R(k) + 2*10^floor(k/2-1) is prime, where R(k) = (10^k-1)/9 (repunit: A002275).

%C The corresponding primes are near-repunit primes, cf. A105992.

%C In base 10, R(k) + 2*10^floor(k/2-1) has ceiling(k/2) digits 1, one digit 3 and again floor(k/2-1) digits 1: for even as well as odd k, there is a digit 3 just left of the middle of the repunit of length k.

%C No term can be equivalent to 1 (mod 3). - _Chai Wah Wu_, Feb 07 2020

%H Brady Haran and Simon Pampena, <a href="https://www.youtube.com/watch?v=HPfAnX5blO0">Glitch Primes and Cyclops Numbers</a>, Numberphile video (2015).

%e For k = 2, R(2) + 2*10^(1-1) = 13 is prime.

%e For k = 3, R(3) + 2*10^(1-1) = 113 is prime.

%e For k = 5, R(5) + 2*10^(2-1) = 11131 is prime.

%e For k = 8, R(8) + 2*10^(4-1) = 11113111 is prime.

%o (PARI) for(n=2,999,isprime(p=10^n\9+2*10^(n\2-1))&&print1(n","))

%Y Cf. A105992 (near-repunit primes), A002275 (repunits), A011557 (powers of 10).

%Y Cf. A331865 (variant with floor(n/2) instead of floor(n/2-1)), A331860, A331863 (variants with digit 2 resp. 0 instead of digit 3).

%K nonn,base,hard,more

%O 1,1

%A _M. F. Hasler_, Jan 30 2020

%E a(13)-a(16) from _Daniel Suteu_, Feb 01 2020

%E a(17) from _Michael S. Branicky_, Feb 03 2023