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

2,1

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

Links

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

Example

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.

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}]

Crossrefs

Cf. A023037, A088790.

Sequence in context: A222762 A290376 A089711 * A173759 A001039 A357855

Adjacent sequences: A374170 A374171 A374172 * A374174 A374175 A374176

Keyword

base,nonn,changed

Author

Robert P. P. McKone, Jun 30 2024

Status

approved