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A107123 Numbers k such that (10^(2*k+1)+18*10^k-1)/9 is prime.

%I A107123 #57 Jul 31 2025 19:12:33
%S A107123 0,1,2,19,97,9818
%N A107123 Numbers k such that (10^(2*k+1)+18*10^k-1)/9 is prime.
%C A107123 A number k is in the sequence iff the palindromic number 1(k).3.1(k) is prime (1(k) means k copies of 1; dot between numbers means concatenation). If k is a positive term of the sequence then k is not of the form 3m, 6m+4, 12m+10, 28m+5, 28m+8, etc. (the proof is easy).
%C A107123 The palindromic number 1(k).2.1(k) is never prime for k > 0 because it is (1.0(k-1).1)*(1(k+1)). - _Robert Israel_, Jun 11 2015
%C A107123 a(7) > 10^5. - _Robert Price_, Apr 02 2016
%D A107123 C. Caldwell and H. Dubner, "Journal of Recreational Mathematics", Volume 28, No. 1, 1996-97, pp. 1-9.
%H A107123 Patrick De Geest, World!Of Numbers, <a href="http://www.worldofnumbers.com/wing.htm#pwp131">Palindromic Wing Primes (PWP's)</a>
%H A107123 Makoto Kamada, <a href="https://stdkmd.net/nrr/1/11311.htm#prime">Prime numbers of the form 11...11311...11</a>
%H A107123 <a href="/index/Pri#Pri_rep">Index entries for primes involving repunits</a>.
%F A107123 a(n) = (A077779(n-1)-1)/2, for n > 1. [Corrected by _M. F. Hasler_, Feb 06 2020]
%e A107123 19 is in the sequence because the palindromic number (10^(2*19+1)+18*10^19-1)/9 = 1(19).3.1(19) = 111111111111111111131111111111111111111 is prime.
%p A107123 select(n -> isprime((10^(2*n+1)+18*10^n-1)/9), [$0..100]); # _Robert Israel_, Jun 11 2015
%t A107123 Do[If[PrimeQ[(10^(2n + 1) + 18*10^n - 1)/9], Print[n]], {n, 2500}]
%o A107123 (PARI) for(n=0,1e4,if(ispseudoprime(t=(10^(2*n+1)+18*10^n)\9),print1(t", "))) \\ _Charles R Greathouse IV_, Jul 15 2011
%Y A107123 Cf. A004023, A077775-A077798, A107124, A107125, A107126, A107127, A107648, A107649, A115073, A183174-A183187.
%K A107123 nonn,base,more
%O A107123 1,3
%A A107123 _Farideh Firoozbakht_, May 19 2005
%E A107123 Edited by _Ray Chandler_, Dec 28 2010