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A241100 Smallest prime with length n having at least n-1 identical digits.

%I A241100 #27 Oct 23 2024 16:04:24
%S A241100 2,11,101,1117,10111,101111,1111151,11110111,101111111,1111111121,
%T A241100 11111111113,101111111111,1111111118111,11111111111411,
%U A241100 111111111116111,1111111111111181,11111111101111111,101111111111111111,1111111111111111111,11011111111111111111
%N A241100 Smallest prime with length n having at least n-1 identical digits.
%C A241100 Conjecture: each term consists of at least n-1 digits 1. - _Chai Wah Wu_, Dec 10 2015
%C A241100 From _Robert G. Wilson v_, Dec 14 2015: (Start)
%C A241100 Terms for which the digit d is the other digit besides the 1's:
%C A241100 d:
%C A241100 0: 3, 5, 6, 8, 9, 12, 17, 18, 20, 24, 26, 29, 30, 32, 33, 35, 36, 38, 39, 42, ..., ; n cannot be congruent to 1 (mod 3);
%C A241100 1: 2, 19, 23, not 317, nor 1031, ..., (see A004023); n cannot be congruent to 0 (mod 3)
%C A241100 2: 1, 10, 34, 46, 67, 75, 100, 103, 142, 148, 154, 175, 198, 232, 244, 274, ..., ;
%C A241100 3: 11, 63, 69, 71, 87, 123, 125, 165, 191, 197, 203, 239, 254, 255, 275, 279, ..., ;
%C A241100 4: 14, 31, 55, 76, 85, 91, 95, 109, 121, 127, 130, 143, 155, 163, 166, 178, ..., ;
%C A241100 5: 7, 22, 28, 37, 45, 52, 60, 94, 111, 132, 133, 139, 159, 160, 172, 184, ..., ;
%C A241100 6: 15, 41, 57, 59, 135, 156, 171, 213, 311, 336, 339, 345, 347, 350, 431, ..., ;
%C A241100 7: 4, 40, 47, 58, 64, 70, 101, 106, 112, 115, 118, 131, 136, 145, 157, 169, ..., ;
%C A241100 8: 13, 16, 25, 43, 49, 61, 73, 79, 82, 88, 93, 97, 99, 117, 124, 141, 151, ..., ;
%C A241100 9: 21, 27, 65, 81, 119, 167, 179, 183, 189, 237, 242, 287, 299, 333, 356, ..., . (End)
%H A241100 Chai Wah Wu, <a href="/A241100/b241100.txt">Table of n, a(n) for n = 1..1000</a>
%p A241100 with(numtheory):lst:={}:nn:=80:kk:=0:T:=array(1..nn):U:=array(1..20):
%p A241100    for n from 2 to nn do:
%p A241100      for i from 1 to n do:
%p A241100      T[i]:=1:
%p A241100      od:
%p A241100      ii:=0:
%p A241100       for k from 0 to 9 while(ii=0)do:
%p A241100         for j from 1 to n while(ii=0)do:
%p A241100         T[j]:=k:s:=sum('T[i]*10^(n-i)', 'i'=1..n):
%p A241100          if type(s,prime)=true and length(s)=n
%p A241100          then
%p A241100          ii:=1: kk:=kk+1:U[kk]:=s:
%p A241100          else
%p A241100          T[j]:=1:
%p A241100          fi:
%p A241100        od:
%p A241100      od:
%p A241100     od :
%p A241100      print(U) :
%t A241100 f[n_] := Block[{k = n - 2, p = 0, r = (10^n - 1)/9, s}, If[ Mod[n, 3] != 1, While[p = r - 10^k; k > 0 && ! PrimeQ@ p, k--]]; If[ Mod[p, 10] == 0, k = 0; s = Select[Range[0, 8], Mod[# + n, 3] > 0 &]; While[p = Select[r + 10^k*s, PrimeQ]; k < n && p == {}, k++]]; p = Min@ p]; Array[f, 20] (* _Robert G. Wilson v_, Dec 14 2015 *)
%t A241100 Table[SelectFirst[Sort[Flatten[Table[Select[FromDigits/@Permutations[PadRight[{d},n,1]],IntegerLength[#] == n&],{d,0,9}]]],PrimeQ],{n,20}] (* Assumes that Chai Wah Wu's conjecture, above, is correct. *) (* _Harvey P. Dale_, Oct 23 2024 *)
%o A241100 (Python)
%o A241100 from __future__ import division
%o A241100 from sympy import isprime
%o A241100 def A241100(n):
%o A241100     for i in range(1,10):
%o A241100         x = i*(10**n-1)//9
%o A241100         for j in range(n-1,-1,-1):
%o A241100             for k in range(i,-1,-1):
%o A241100                 if j < n-1 or k < i:
%o A241100                     y = x-k*(10**j)
%o A241100                     if isprime(y):
%o A241100                         return y
%o A241100         for j in range(n):
%o A241100             for k in range(1,9-i+1):
%o A241100                 y = x+k*(10**j)
%o A241100                 if isprime(y):
%o A241100                     return y # _Chai Wah Wu_, Dec 29 2015
%K A241100 nonn,base
%O A241100 1,1
%A A241100 _Michel Lagneau_, Apr 16 2014
%E A241100 a(4), a(7), a(10), a(11), a(13)-a(16) corrected by _Chai Wah Wu_, Dec 10 2015
%E A241100 a(1) from _Robert G. Wilson v_, Dec 11 2015