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A286342 Smallest beastly prime in base n: smallest prime p with a base-n expansion containing the substring 666.

%I A286342 #19 Aug 17 2022 11:07:40
%S A286342 2399,3511,4919,6661,2129,11311,14281,17729,21701,26209,26407,37049,
%T A286342 43441,50527,252823,66931,64153,86561,19531,109673,122651,136601,
%U A286342 151561,167593,184703,202949,222361,242971,50441,287933,261707,338137,365291,393847,79259
%N A286342 Smallest beastly prime in base n: smallest prime p with a base-n expansion containing the substring 666.
%C A286342 No such p exists for n < 7.
%C A286342 Does p exist for every n > 6?
%C A286342 Answer: yes. For a given n, consider the sequence {k*n^4 + 6*n^3 + 6*n^2 + 6*n + 1}. By Dirichlet's theorem on arithmetic progressions, there exist infinitely many primes of this form, and they all end in 6661 in base n. - _Jianing Song_, Feb 03 2019
%H A286342 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions">Dirichlet's theorem on arithmetic progressions</a>
%F A286342 n^3 < a(n) << n^20. - _Charles R Greathouse IV_, May 13 2017
%F A286342 Probably n^3 < a(n) < n^4 for all but finitely many n. It appears the only exceptions are 21 and 52. If there are any others they are larger than 10^7; the expected number of larger exceptions is about 10^-89814. - _Charles R Greathouse IV_, May 13 2017
%e A286342 For n = 7: 2399 written in base 7 is 6665. Since 2399 is the smallest prime that contains the substring 666 in its base-7 expansion, a(7) = 2399.
%t A286342 Table[k = FromDigits[#, b]; While[Nand[PrimeQ@ k, Length@ SequencePosition[IntegerDigits[k, b], #] > 0], k++]; k, {b, 7, 41}] &@ ConstantArray[6, 3] (* _Michael De Vlieger_, May 08 2017 *)
%o A286342 (PARI) a(n) = forprime(p=1, , my(subs=[6, 6, 6], dbn=digits(p, n)); for(k=1, #dbn-2, my(v=[dbn[k], dbn[k+1], dbn[k+2]]); if(v==subs, return(p))))
%Y A286342 Cf. A131645.
%K A286342 nonn,base,less
%O A286342 7,1
%A A286342 _Felix Fröhlich_, May 07 2017