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 A034388 Smallest prime containing at least n consecutive identical digits. 23
 2, 11, 1117, 11113, 111119, 2999999, 11111117, 111111113, 1777777777, 11111111113, 311111111111, 2111111111111, 17777777777777, 222222222222227, 1333333333333333, 11111111111111119, 222222222222222221, 1111111111111111111, 1111111111111111111 (list; graph; refs; listen; history; text; internal format)
 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 EXTENSIONS Edited by M. F. Hasler, Feb 25 2016 Edited by Robert Israel, Feb 26 2016 STATUS approved

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Last modified December 9 17:01 EST 2022. Contains 358701 sequences. (Running on oeis4.)