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

7,1

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

Links

Table of n, a(n) for n=7..41.

Wikipedia, Dirichlet's theorem on arithmetic progressions

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

Example

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.

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 *)

Prog

(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))))

Crossrefs

Cf. A131645.

Sequence in context: A323344 A323341 A108055 * A204357 A043420 A233880

Adjacent sequences: A286339 A286340 A286341 * A286343 A286344 A286345

Keyword

nonn,base,less

Author

Felix Fröhlich, May 07 2017

Status

approved