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

%I A374173 #32 Jan 02 2025 09:35:04
%S A374173 3,13,683,3907,55987,960803,19173967,435848051,11111111113,
%T A374173 1540683021299,19453310068921,328114698808283,45302797058044219,
%U A374173 469172025408063623,19676527011956855059,878942778254232811943,120353718818554114936591,109912203092239643840221
%N A374173 a(n) is the smallest prime whose base-n representation contains a run of at least n identical digits.
%C A374173 a(2) to a(18) are all increasing, but a(19) is smaller than a(18).
%C A374173 a(n) = A023037(n) for n in A088790. - _Robert Israel_, Dec 31 2024
%H A374173 Robert Israel, <a href="/A374173/b374173.txt">Table of n, a(n) for n = 2..385</a>
%e A374173 a(2) = 3 = 11_2.
%e A374173 a(3) = 13 = 111_3.
%e A374173 a(11) = 1540683021299 = 544444444444_11.
%e A374173 a(18) = 120353718818554114936591 = 3111111111111111111_18.
%e A374173 a(19) = 109912203092239643840221 = 1111111111111111111_19.
%p A374173 f:= proc(n) local t,Q,i,j;
%p A374173   t:= (n^n-1)/(n-1);
%p A374173   if isprime(t) then return t fi;
%p A374173   for i from 1 to n-1 do
%p A374173     Q:= select(isprime, [seq(i*t*n+j,j=1..n-1),
%p A374173          seq(i*n^n+j*t,j=1..n-1)]);
%p A374173     if Q <> [] then return min(Q) fi;
%p A374173   od;
%p A374173   FAIL
%p A374173 end proc:
%p A374173 map(f, [$2..20]); # _Robert Israel_, Dec 31 2024
%t A374173 d[n_]:=d[n]=Table[Table[m,n],{m,0,n-1}];
%t A374173 dpre[n_]:=Flatten[Table[{m}~Join~#&/@d[n],{m,0,n-1}],1];
%t A374173 dpost[n_]:=Flatten[Table[Map[#~Join~{m}&,d[n]],{m,0,n-1}],1];
%t A374173 dprepost[n_]:=Flatten[Table[Map[{j}~Join~#~Join~{m}&,d[n]],{m,0,n-1},{j,0,n-1}],2];
%t A374173 c[n_]:=c[n]=DeleteDuplicates[Sort[Select[FromDigits[#,n]&/@Join[d[n],dpre[n],dpost[n],dprepost[n]],#>n&]]];
%t A374173 a[n_]:=a[n]=Do[If[PrimeQ[q],Return[q];Break[];],{q,c[n]}];
%t A374173 Table[a[n],{n,2,19}]
%Y A374173 Cf. A023037, A088790.
%K A374173 base,nonn
%O A374173 2,1
%A A374173 _Robert P. P. McKone_, Jun 30 2024