A241206 Greatest n-digit prime having at least n-1 identical digits.

7, 97, 997, 9949, 99991, 999979, 9999991, 99999989, 999999929, 9999999929, 99999999599, 999999999989, 9999999999799, 99999999999959, 999999999999989, 9999999999999199, 99999999999999997, 999999999999999989, 9999999999999999919, 99999999999999999989, 999999999999999999899, 9999999999999999999929

Offset: 1

Author

Michel Lagneau, Apr 17 2014

Keywords

• nonn
• base

Comments

Not the same as A069661 (Smallest n-digit prime with maximum digit sum). For example, A069661(10) = 9899989999 with only n-2 = 8 identical digits.

Conjecture: each term consists of at least n-1 digits 9 and no digit 0. - Chai Wah Wu, Dec 10 2015

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..1000 (corrected by _Georg Fischer_, Jan 20 2019)

Crossrefs

Cf. A069661, A241100, A178007.

Programs

Maple

with(numtheory):lst:={}:nn:=30:kk:=0:T:=array(1..nn):U:=array(1..20):
   for n from 2 to nn do:
     for i from 1 to n do:
     T[i]:=9:
     od:
     ii:=0:
        for j from 1 to n while(ii=0)do:
        for k from 9 by -1 to 0 while(ii=0)do:
        T[n-j+1]:=k:s:=sum('T[i]*10^(n-i)', 'i'=1..n):
         if type(s,prime)=true and length(s)=n
         then
         ii:=1: kk:=kk+1:U[kk]:=s:
         else
         T[n-j+1]:=9:
         fi:
       od:
     od:
    od :
     print(U) :

Mathematica

Table[SelectFirst[Reverse@ Prime@ Range[PrimePi[10^(n - 1)] + 1, PrimePi[10^n - 1]], Max@ DigitCount@ # >= (n - 1) &], {n, 2, 8}]
(* WARNING: the following assumes the conjecture is true WARNING *)
Table[SelectFirst[Select[Reverse@ Union@ Map[FromDigits, Join @@ Map[Permutations[Append[Table[9, {n - 1}], #]] &, Range[0, 9]]], PrimeQ@ # && IntegerLength@ # == n &], Max@ DigitCount@ # >= (n - 1) &], {n, 2, 20}] (* Michael De Vlieger, Dec 10 2015, Version 10 *)

Python

from _future_ import division
from sympy import isprime
def A241206(n):
    for i in range(9,0,-1):
        x = i*(10**n-1)//9
        for j in range(n-1,-1,-1):
            for k in range(9-i,-1,-1):
                y = x + k*(10**j)
                if isprime(y):
                    return y
        for j in range(n):
            for k in range(1,i+1):
                if j < n-1 or k < i:
                    y = x-k*(10**j)
                    if isprime(y):
                        return y # Chai Wah Wu, Dec 29 2015

Extensions

a(1) added - N. J. A. Sloane, Dec 29 2015