A065586 -id:A065586 - cp's OEIS frontend

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

Erich Friedman and N. J. A. Sloane

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

			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).

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.

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

Cf. A065821, A065586, A084673.

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

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

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

Edited by M. F. Hasler, Feb 25 2016

Edited by Robert Israel, Feb 26 2016

A037059 Smallest prime containing exactly n 3's.

Original entry on oeis.org

2, 3, 233, 2333, 23333, 313333, 3233333, 31333333, 333233333, 3233333333, 23333333333, 333313333333, 3333333333383, 33133333333333, 323333333333333, 1333333333333333, 23333333333333333, 333333133333333333, 3333313333333333333, 33313333333333333333

Offset: 0

Patrick De Geest, Jan 04 1999

For almost all n >= 0, a(n) equals [10^(n+1)/3] with one of the (first) digits 3 replaced by a digit 1 or 2. We conjecture that in the few other cases (e.g., for n = 12, 119, ...) the statement holds with some digit 3 replaced by a digit among {4, 5, 7, 8}, except for the special case a(1) = 3. - M. F. Hasler, Feb 22 2016

M. F. Hasler, Table of n, a(n) for n = 0..200; a(1) through a(100) from Harvey P. Dale.

Different from A065580.
Cf. A065586, A065580, A037058, A034388, A036507-A036536.
Cf. A037053, A037055, A037057, A037061, A037063, A037065, A037067, A037069, A037071.

f[n_, b_] := Block[{k = 10^(n + 1), p = Permutations[ Join[ Table[b, {i, 1, n}], {x}]], c = Complement[Table[j, {j, 0, 9}], {b}], q = {}}, Do[q = Append[q, Replace[p, x -> c[[i]], 2]], {i, 1, 9}]; r = Min[ Select[ FromDigits /@ Flatten[q, 1], PrimeQ[ # ] & ]]; If[r ? Infinity, r, p = Permutations[ Join[ Table[ b, {i, 1, n}], {x, y}]]; q = {}; Do[q = Append[q, Replace[p, {x -> c[[i]], y -> c[[j]]}, 2]], {i, 1, 9}, {j, 1, 9}]; Min[ Select[ FromDigits /@ Flatten[q, 1], PrimeQ[ # ] & ]]]]; Table[ f[n, 3], {n, 1, 18}]
Table[Sort[Flatten[Table[Select[FromDigits/@Permutations[Join[{n},PadRight[{},i,3]]], PrimeQ],{n,0,9}]]][[1]],{i,20}] (* Harvey P. Dale, Feb 28 2015 *)

A037059(n)={if(n==1,3,my(t=10^(n+1)\3); forvec(v=[[-1, n], [-2, -1]], ispseudoprime(p=t+10^(n-v[1])*v[2]) && return(p)); forvec(v=[[0, n], [1, 5]], ispseudoprime(p=t+10^v[1]*v[2]) && return(p)))} \\ M. F. Hasler, Feb 22 2016

a(n) = prime(A037058(n)). - Amiram Eldar, Jul 21 2025

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 23 2003

More terms and a(0) = 2 prefixed by M. F. Hasler, Feb 22 2016

A065580 Smallest prime ending in exactly n 3's.

Original entry on oeis.org

3, 233, 2333, 23333, 733333, 10333333, 83333333, 1033333333, 17333333333, 23333333333, 2633333333333, 10333333333333, 53333333333333, 3233333333333333, 1333333333333333, 23333333333333333, 2033333333333333333, 10333333333333333333, 173333333333333333333, 1733333333333333333333

Offset: 1

Robert G. Wilson v, Nov 28 2001

Harry J. Smith, Table of n, a(n) for n = 1..100

Cf. A037059, A065586, A065821, A065581, A065582.

Do[a = Table[3, {n} ]; k = 0; While[ b = FromDigits[ Join[ IntegerDigits[k], a]]; Mod[k, 10] == 3 || !PrimeQ[b], k++ ]; Print[b], {n, 1, 17} ]

a(n)={ my(t=10^n, b=(t-1)/3, d=0); while (!isprime(b + t*d), d++; if(d%10==3, d++)); b + t*d } \\ Harry J. Smith, Oct 23 2009

A065589 Smallest prime beginning with exactly n 6's.

Original entry on oeis.org

2, 61, 661, 6661, 666607, 666667, 66666629, 66666667, 666666667, 6666666661, 66666666667, 6666666666629, 66666666666629, 666666666666631, 66666666666666047, 66666666666666601, 6666666666666666059, 666666666666666661, 66666666666666666601, 66666666666666666667

Offset: 0

Robert G. Wilson v, Nov 28 2001

David A. Corneth, Table of n, a(n) for n = 0..700 (first 201 terms from M. F. Hasler)

Cf. A037065, A065584, A065585, A065586, A065587, A065588, (this sequence), A065590, A065591, A065592.

Do[a = Table[6, {n}]; k = 0; While[b = FromDigits[ Join[a, IntegerDigits[k] ]]; First[ IntegerDigits[k]] == 6 || !PrimeQ[b], k++ ]; Print[b], {n, 1, 17} ]

a(n) = {if(n==0, return(2)); my(cs = 60*(10^n\9), pow10 = 10); for(i = 1, oo, np = cs; d = 0; while(d < pow10, np = nextprime(np + 1); d = np - cs; if(d < pow10 && digits(d)[1] != 6 || 10*d < pow10, return(np))); cs*=10; pow10*=10)} \\ David A. Corneth, Sep 06 2023

Corrected by Don Reble, Jan 17 2007

Offset corrected by Sean A. Irvine, Sep 06 2023

A123568 Prime numbers of the form (10^n - 7)/3.

Original entry on oeis.org

31, 331, 3331, 33331, 333331, 3333331, 33333331, 333333333333333331, 3333333333333333333333333333333333333331, 33333333333333333333333333333333333333333333333331

Offset: 1

Artur Jasinski, Nov 12 2006

nonn

The number of initial 3s is n - 1.

Note that each n from 2 to 8 gives primes, but after that the n that correspond to primes are progressively further apart. Singh (1997) gives this as an example of why mathematicians don't trust a preponderance of evidence as proof: in the 17th century, when factoring numbers with as few as eight digits wasn't as easy as it is today, the pattern suggested that all numbers of this form are prime. - Alonso del Arte, Nov 11 2012

			a(7) = 33333331 because that is the seventh number of the specified form to be prime.
333333331 is not in the sequence because it is composite, being the product of 17 and 19607843.

Simon Singh, Fermat's Enigma. New York: Walker & Company (1997) p. 159.

Alois P. Heinz, Table of n, a(n) for n = 1..17

Cf. A055557, A051200, A033175, A065586, A068104.

Do[If[PrimeQ[(10^n - 7)/3], Print[(10^n - 7)/3]], {n, 1, 100}] (* Jasinski *)
Select[(10^Range[50] - 7)/3, PrimeQ[#] &] (* Alonso del Arte, Nov 11 2012 *)
Select[Table[FromDigits[PadLeft[{1},n,3]],{n,50}],PrimeQ] (* Harvey P. Dale, Dec 05 2018 *)

select(ispseudoprime, vector(20, n, (10^n-7)/3)) \\ Charles R Greathouse IV, Nov 12 2012

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A034388 Smallest prime containing at least n consecutive identical digits.

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A037059 Smallest prime containing exactly n 3's.

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A065580 Smallest prime ending in exactly n 3's.

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A065589 Smallest prime beginning with exactly n 6's.

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A123568 Prime numbers of the form (10^n - 7)/3.

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