A037067 Smallest prime containing exactly n 7's.

2, 7, 277, 1777, 47777, 727777, 7477777, 77767777, 577777777, 1777777777, 67777777777, 377777777777, 7177777777777, 17777777777777, 577777777777777, 2777777777777777, 77777767777777777, 377777777777777777, 2777777777777777777, 71777777777777777777

Offset: 0

Author

Patrick De Geest, Jan 04 1999

Keywords

• nonn
• base

Comments

We conjecture that for all n >= 2, a(n) equals floor(10^(n+1)/9)*7 with one of the (first) digits 7 replaced by a digit among {0, ..., 6}. - M. F. Hasler, Feb 22 2016

The conjecture is false: a(668) = 7*(10^669-1)/9 + 10^276. - Robert Israel, Jul 13 2016

Links

• M. F. Hasler and Robert Israel, Table of n, a(n) for n = 0..998 (n = 0..200 from M. F. Hasler)

Crossrefs

Cf. A065590, A065581, A037066, A034388, A036507-A036536.

Cf. A037053, A037055, A037057, A037059, A037061, A037063, A037065, A037069, A037071.

Programs

Maple

F:= proc(n) local x0,i,j;
  x0:= 7/9*(10^(n+1)-1);
  for j from 1 to 6 do
    if isprime(x0 + (j-7)*10^n) then
      return x0 + (j-7)*10^n fi od;
  for i from n-1 to 0 by -1 do
    for j from 0 to 6 do
     if isprime(x0 + (j-7)*10^i) then
       return x0 + (j-7)*10^i fi od od;
  for i from 0 to n do
    for j from 8 to 9 do
       if isprime(x0 + (j-7)*10^i) then
         return x0 + (j-7)*10^i fi
  od od:
end proc:
F(0):= 2: F(1):= 7:
map(F, [$0..100]); # Robert Israel, Jul 13 2016

Mathematica

f[n_, b_] := Block[{k = 10^(n + 1), p = Permutations[ Join[ Table[b, {i, 1, n}], {x}]], c = Complement[Table[j, {j, 0, 9}], {b}], q = {}}, Do[q = Append[q, Replace[p, x -> c[[i]], 2]], {i, 1, 9}]; r = Min[ Select[ FromDigits /@ Flatten[q, 1], PrimeQ[ # ] & ]]; If[r ? Infinity, r, p = Permutations[ Join[ Table[ b, {i, 1, n}], {x, y}]]; q = {}; Do[q = Append[q, Replace[p, {x -> c[[i]], y -> c[[j]]}, 2]], {i, 1, 9}, {j, 1, 9}]; Min[ Select[ FromDigits /@ Flatten[q, 1], PrimeQ[ # ] & ]]]]; Table[ f[n, 7], {n, 1, 18}]

PARI

A037067(n)={my(t=10^(n+1)\9*7); forvec(v=[[-1, n], [-7, -1]], ispseudoprime(p=t+10^(n-v[1])*v[2]) && return(p)); error} \\ M. F. Hasler, Feb 22 2016

Formula

a(n) = prime(A037066(n)). - Amiram Eldar, Jul 21 2025

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 23 2003

More terms from and a(0) = 2 prepended by M. F. Hasler, Feb 22 2016