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%I #36 Feb 09 2020 03:01:03
%S 3,26,186,206,258,3486,12602
%N Numbers n for which R(n) - 10^floor(n/2) is prime, where R(n) = (10^n-1)/9.
%C The corresponding primes are a subsequence of A065074: near-repunit primes that contain the digit 0.
%C In base 10, R(n) - 10^floor(n/2) has ceiling(n/2)-1 digits 1, one digit 0, and again floor(n/2) digits 1. For odd n, this is a palindrome, for even n the digit 0 is just left to the middle of the number.
%C There can't be an odd term > 3 because the corresponding palindrome factors as R((n-1)/2)*(10^((n+1)/2) + 1).
%C No term can be congruent 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 n = 3, R(n) - 10^floor(n/2) = 101 is prime.
%e For n = 26, R(n) - 10^floor(n/2) = 11111111111101111111111111 is prime.
%o (PARI) for(n=0,9999,isprime(p=10^n\9-10^(n\2))&&print1(n","))
%Y Cf. A002275 (repunits), A004023 (indices of prime repunits), A011557 (powers of 10), A065074 (near-repunit primes that contain the digit 0), A105992 (near-repunit primes), A138148 (Cyclops numbers with digits 0 & 1).
%Y Cf. A331860 (variant with digit 2 instead of digit 0), A331863 (variant with floor(n/2-1) instead of floor(n/2)).
%K nonn,base,hard,more
%O 1,1
%A _M. F. Hasler_, Jan 30 2020
%E a(6)-a(7) from _Giovanni Resta_, Jan 31 2020