A037055 Smallest prime containing exactly n 1's.

2, 13, 11, 1117, 10111, 101111, 1111151, 11110111, 101111111, 1111111121, 11111111113, 101111111111, 1111111118111, 11111111111411, 111111111116111, 1111111111111181, 11111111101111111, 101111111111111111, 1111111111111111171, 1111111111111111111, 111111111111111119111

Offset: 0

Author

Patrick De Geest, Jan 04 1999

Keywords

• nonn
• base

Comments

For n > 1, A037055 is conjectured to be identical to A084673. - Robert G. Wilson v, Jul 04 2003

a(n) = A002275(n) for n in A004023. For all other n < 900, a(n) has n+1 digits. - Robert Israel, Feb 21 2016

Links

• Robert Israel, Table of n, a(n) for n = 0..900

Crossrefs

Cf. A084673, A065584, A065821, A037054, A034388, A036507-A036536.

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

Programs

Maple

f:= proc(n) local m,d,r,x;
   r:= (10^n-1)/9;
   if isprime(r) then return r fi;
   r:= (10^(n+1)-1)/9;
   for m from n-1 to 1 by -1 do
     x:= r - 10^m;
     if isprime(x) then return x fi;
   od;
   for m from 0 to n do
     for d from 1 to 8 do
        x:= r + d*10^m;
        if isprime(x) then return x fi;
     od
   od;
   error("Needs more than n+1 digits")
end proc:
map(f, [$0..100]); # Robert Israel, Feb 21 2016

Mathematica

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, 1], {n, 1, 18}]
Join[{2, 13}, Table[Sort[Flatten[Table[Select[FromDigits/@Permutations[Join[{n}, PadRight[{}, i, 1]]], PrimeQ], {n, 0, 9}]]][[1]], {i, 2, 20}]] (* Vincenzo Librandi, May 11 2017 *)

PARI

A037055(n)={my(p,t=10^(n+1)\9); forstep(k=n+1,1,-1, ispseudoprime(p=t-10^k) && return(p)); forvec(v=[[0, n], [1, 8]], ispseudoprime(p=t+10^v[1]*v[2]) && return(p))} \\ M. F. Hasler, Feb 22 2016

Formula

a(n) = the smallest prime in { R-10^n, R-10^(n-1), ..., R-10; R+a*10^b, a=1, ..., 8, b=0, 1, 2, ..., n }, where R = (10^(n+1)-1)/9 is the (n+1)-digit repunit. - M. F. Hasler, Feb 25 2016

a(n) = prime(A037054(n)). - Amiram Eldar, Jul 21 2025

Extensions

More terms from Sascha Kurz, Feb 10 2003

Edited by Robert G. Wilson v, Jul 04 2003

a(0) = 2 inserted by Robert Israel, Feb 21 2016