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%I #32 Jul 23 2024 08:49:42
%S 6,7,12,31,58,127,454,556,558,604,2944,8118,12078,16942,26268,45198
%N Numbers k such that R(k) + 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) + 10^floor(k/2-1) has ceiling(k/2) digits 1, one digit 2 and again floor(k/2-1) digits 1: for even as well as odd k, there is a digit 2 just left of the middle of the repunit of length k.
%C No term can be congruent to 2 (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 n = 6, R(6) + 10^(3-1) = 111211 is prime.
%e For n = 7, R(7) + 10^(3-1) = 1111211 is prime.
%e For n = 12, R(12) + 10^(6-1) = 111111211111 is prime.
%o (PARI) for(n=2,999,isprime(p=10^n\9+10^(n\2-1))&&print1(n","))
%Y Cf. A105992 (near-repunit primes), A002275 (repunits), A011557 (powers of 10).
%Y Cf. A331861 (variant with floor(n/2) instead of floor(n/2-1)), A331863 (variant with - (digit 0) instead of + (digit 2)).
%K nonn,base,hard,more
%O 1,1
%A _M. F. Hasler_, Jan 30 2020
%E a(8)-a(14) from _Giovanni Resta_, Jan 31 2020
%E a(15)-a(16) from _Michael S. Branicky_, Jul 23 2024