A374173 a(n) is the smallest prime whose base-n representation contains a run of at least n identical digits.

3, 13, 683, 3907, 55987, 960803, 19173967, 435848051, 11111111113, 1540683021299, 19453310068921, 328114698808283, 45302797058044219, 469172025408063623, 19676527011956855059, 878942778254232811943, 120353718818554114936591, 109912203092239643840221

Offset: 2

Author

Robert P. P. McKone, Jun 30 2024

Keywords

• base
• nonn

Comments

a(2) to a(18) are all increasing, but a(19) is smaller than a(18).

a(n) = A023037(n) for n in A088790. - Robert Israel, Dec 31 2024

Examples

			a(2) = 3 = 11_2.
a(3) = 13 = 111_3.
a(11) = 1540683021299 = 544444444444_11.
a(18) = 120353718818554114936591 = 3111111111111111111_18.
a(19) = 109912203092239643840221 = 1111111111111111111_19.
		

Links

• Robert Israel, Table of n, a(n) for n = 2..385

Crossrefs

Cf. A023037, A088790.

Programs

Maple

f:= proc(n) local t,Q,i,j;
  t:= (n^n-1)/(n-1);
  if isprime(t) then return t fi;
  for i from 1 to n-1 do
    Q:= select(isprime, [seq(i*t*n+j,j=1..n-1),
         seq(i*n^n+j*t,j=1..n-1)]);
    if Q <> [] then return min(Q) fi;
  od;
  FAIL
end proc:
map(f, [$2..20]); # Robert Israel, Dec 31 2024

Mathematica

d[n_]:=d[n]=Table[Table[m,n],{m,0,n-1}];
dpre[n_]:=Flatten[Table[{m}~Join~#&/@d[n],{m,0,n-1}],1];
dpost[n_]:=Flatten[Table[Map[#~Join~{m}&,d[n]],{m,0,n-1}],1];
dprepost[n_]:=Flatten[Table[Map[{j}~Join~#~Join~{m}&,d[n]],{m,0,n-1},{j,0,n-1}],2];
c[n_]:=c[n]=DeleteDuplicates[Sort[Select[FromDigits[#,n]&/@Join[d[n],dpre[n],dpost[n],dprepost[n]],#>n&]]];
a[n_]:=a[n]=Do[If[PrimeQ[q],Return[q];Break[];],{q,c[n]}];
Table[a[n],{n,2,19}]