A034388 Smallest prime containing at least n consecutive identical digits.

2, 11, 1117, 11113, 111119, 2999999, 11111117, 111111113, 1777777777, 11111111113, 311111111111, 2111111111111, 17777777777777, 222222222222227, 1333333333333333, 11111111111111119, 222222222222222221, 1111111111111111111, 1111111111111111111

Offset

1,1

Comments

For n in A004023, a(n) = A002275(n). For all other n > 1, a(n) has at least n+1 digits and is (for small n) often of the form a*10^n + b*(10^n-1)/9 or a*(10^n-1)/9*10 + b, with 1 <= a <= 9 and b in {1, 3, 7, 9}. However, for n = 24, 46, 48, 58, 60, 64, 66, ..., more digits are required. Only then a(n) can have a digit 0, and if it has, '0' is often the repeated digit. The first indices where a(n) has more than n+2 digits are n = 208, 277, 346, ... - M. F. Hasler, Feb 25 2016; corrected by Robert Israel, Feb 26 2016

Links

M. F. Hasler and Robert Israel, Table of n, a(n) for n = 1..997 (1..200 from M. F. Hasler)

Chai Wah Wu, Can machine learning identify interesting mathematics? An exploration using empirically observed laws, arXiv:1805.07431 [cs.LG], 2018.

Example

a(1) = 2 because this is the smallest prime.
a(2) = 11 because this repunit with n=2 digits is prime.
a(3) = 1117 is the smallest prime with 3 repeated digits.
a(19) = (10^19-1)/9 = R(19) is again a repunit prime. Since all primes with 18 consecutive repeated digits have at least 19 digits, also a(18) = a(19). The same happens for a(22) = a(23).

Maple

f:= proc(n) local d, k, x, y, z, xs, ys, zs, c, cands;
  for d from n do
    cands:= NULL;
    for k from 0 to d-n do
      if k = 0 then zs:= [0] else zs:= [seq(i, i=1..10^k-1, 2)] fi;
      if d=n+k then xs:= [0]; ys:= [$1..9] else xs:= [$10^(d-k-n-1)..10^(d-k-n)-1]; ys:= [$0..9] fi;
      cands:= cands, seq(seq(seq(z + 10^k*y*(10^n-1)/9 + x*10^(k+n), x = xs), y=ys), z=zs);
    od;
    cands:= sort([cands]);
    for c in cands do if isprime(c) then return(c) fi od;
  od
end proc:
map(f, [$1..30]); # Robert Israel, Feb 26 2016

Mathematica

With[{s = Length /@ Split@ IntegerDigits@ # & /@ Prime@ Range[10^6]}, Prime@ Array[FirstPosition[s, #][[1]] &, Max@ Flatten@ s]] (* Michael De Vlieger, Aug 15 2018 *)

Prog

(PARI) A034388(n)={for(d=0, 9, my(L=[], k=0); for(a=0, 10^d-1, a<10^k||k++; L=setunion(L, vector(10-!a, c, [a*10^n+10^n\9*(c-(a>0)), 1])*10^(d-k))); for(i=1, #L, if(L[i][2]>1, L[i][1]+L[i][2]>L[i][1]=nextprime(L[i][1]), ispseudoprime(L[i][1]))&&return(L[i][1])))} \\ M. F. Hasler, Feb 28 2016

Crossrefs

Cf. A037053, A037055, A037057, A037059, A037061, A037063, A037065, A037067, A037069, A037071.

Cf. A065821, A065586, A084673.

Sequence in context: A011825 A306909 A084673 * A131316 A062636 A343900

Adjacent sequences: A034385 A034386 A034387 * A034389 A034390 A034391

Keyword

nonn,base

Author

Erich Friedman and N. J. A. Sloane.

Extensions

Edited by M. F. Hasler, Feb 25 2016

Edited by Robert Israel, Feb 26 2016

Status

approved