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Numbers k for which R(k) + 3*10^floor(k/2) is prime, where R(k) = (10^k-1)/9 (repunit: A002275).
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%I #26 Sep 24 2024 14:14:36

%S 0,2,5,7,8,10,65,91,208,376,586,2744,3089,19378,20246

%N Numbers k for which R(k) + 3*10^floor(k/2) is prime, where R(k) = (10^k-1)/9 (repunit: A002275).

%C The corresponding primes are a subset of the near-repunit primes A105992 (at least when they have k > 2 digits).

%C In base 10, R(k) + 3*10^floor(k/2) has k digits all of which are 1 except for one digit 4 (for k > 0) located in the center (for odd k) or just to the left of it (for even k): i.e., there are ceiling(k/2)-1 digits 1 to the left and floor(k/2) digits 1 to the right of the digit 4. For odd k, this is a palindrome a.k.a. wing prime, cf. A077780, the subsequence of odd terms.

%C a(14) = 19378 was found by _Amiram Eldar_, verified to be the 14th term in collaboration with the author of the sequence and factordb.com. The term a(13) = 3089 corresponds to a certified prime (Ivan Panchenko, 2011, cf. factordb.com); a(12) and a(14) are only PRP as far as we know.

%H Chris Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/xpage/nearrepunitprime.html">Near-repunit prime</a>.

%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 = 0, R(0) + 3*10^floor(0/2) = 3 is prime.

%e For n = 2, R(2) + 3*10^floor(2/2) = 41 is prime.

%e For n = 5, R(5) + 3*10^floor(5/2) = 11411 is prime.

%e For n = 7, R(7) + 3*10^floor(7/2) = 1114111 is prime.

%e For n = 8, R(8) + 3*10^floor(8/2) = 11141111 is prime.

%t Select[Range[0, 2500], PrimeQ[(10^# - 1)/9 + 3*10^Floor[#/2]] &]

%o (PARI) for(n=0,9999,ispseudoprime(p=10^n\9+3*10^(n\2))&&print1(n","))

%Y Cf. A105992 (near-repunit primes), A002275 (repunits), A004023 (indices of prime repunits), A011557 (powers of 10).

%Y Cf. A331862, A331861, A331865, A331869 (variants with digit 0, 2, 3 or 5 instead of 4), A331867 (variant with floor(n/2-1) instead of floor(n/2)).

%Y Cf. A077780 (odd terms).

%K nonn,base,hard,more

%O 1,2

%A _M. F. Hasler_, Jan 30 2020

%E a(15) from _Michael S. Branicky_, Sep 24 2024