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%I #15 Jun 24 2022 04:41:52
%S 2,5,557,5557,155557,555557,15555557,55555553,3555555551,5555555557,
%T 525555555557,555555555551,5555555555551,355555555555559,
%U 555555555555557,51555555555555551,545555555555555551,555555555555555559,15555555555555555557,155555555555555555551
%N Smallest prime containing exactly n 5's.
%C For n > 1, the last digit of n cannot be 5, therefore a(n) must have at least n+1 digits. It is probable that none among [10^n/9]*50 + {1,3,7,9} is prime in which case a(n) must have n+2 digits. We conjecture that for all n > 1, a(n) equals [10^(n+1)/9]*50 + b with 1 <= b <= 9 and one of the (first) digits 5 replaced by a 0, 1, 2, 3 or 4. - _M. F. Hasler_, Feb 22 2016
%H M. F. Hasler, <a href="/A037063/b037063.txt">Table of n, a(n) for n = 0..200</a>
%t 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, 5], {n, 1, 18}]
%o (PARI) A037063(n)={my(p, t=10^(n+1)\9*50); n>1 && forvec(v=[[-1, n], [-5, -1]], nextprime(p=t+10^(n-v[1])*v[2])-p<10 && return(nextprime(p)));1+4^n} \\ _M. F. Hasler_, Feb 22 2016
%Y Cf. A065588, A037062, A034388, A036507-A036536.
%Y Cf. A037053, A037055, A037057, A037059, A037061, A037065, A037067, A037069, A037071.
%K nonn,base
%O 0,1
%A _Patrick De Geest_, Jan 04 1999
%E More terms from _Randall L Rathbun_, Jan 11 2002
%E Edited and corrected by _Robert G. Wilson v_, Jul 04 2003
%E More terms and a(0) = 2 from _M. F. Hasler_, Feb 22 2016
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