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

2399, 3511, 4919, 6661, 2129, 11311, 14281, 17729, 21701, 26209, 26407, 37049, 43441, 50527, 252823, 66931, 64153, 86561, 19531, 109673, 122651, 136601, 151561, 167593, 184703, 202949, 222361, 242971, 50441, 287933, 261707, 338137, 365291, 393847, 79259

Offset: 7

Author

Felix Fröhlich, May 07 2017

Keywords

• nonn
• base
• less

Comments

No such p exists for n < 7.

Does p exist for every n > 6?

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

Examples

			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.
		

Links

• Wikipedia, Dirichlet's theorem on arithmetic progressions

Crossrefs

Cf. A131645.

Programs

Mathematica

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 *)

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))))

Formula

n^3 < a(n) << n^20. - Charles R Greathouse IV, May 13 2017

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