cp's OEIS Frontend

A037069 Smallest prime containing exactly n 8's.

%I A037069 #19 Jul 21 2025 00:22:54
%S A037069 2,83,881,8887,88883,888887,28888883,88888883,888888883,48888888883,
%T A037069 288888888889,888888888887,48888888888883,88888888888889,
%U A037069 888888888888883,18888888888888883,88888888888888889,2888888888888888887,8888888888888888881,388888888888888888889
%N A037069 Smallest prime containing exactly n 8's.
%C A037069 The last digit of n cannot be 8, therefore a(n) must have at least n+1 digits. It is probable (using [] for floor) that none among [10^n/9]*80 + {1,3,7,9} is prime in which case a(n) must have n+2 digits. We conjecture that for all n >= 0, a(n) equals [10^(n+1)/9]*80 + b with 1 <= b <= 9 and one of the (first) digits 8 replaced by a digit among {0, ..., 7}. - _M. F. Hasler_, Feb 22 2016
%H A037069 M. F. Hasler, <a href="/A037069/b037069.txt">Table of n, a(n) for n = 0..200</a>
%F A037069 a(n) = prime(A037068(n)). - _Amiram Eldar_, Jul 21 2025
%t A037069 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, 8], {n, 1, 18}]
%o A037069 (PARI) A037069(n)={my(p, t=10^(n+1)\9*80); forvec(v=[[-1, n], [-8, -1]], nextprime(p=t+10^(n-v[1])*v[2])-p<10 && return(nextprime(p)))} \\ ~
%Y A037069 Cf. A065591, A037068, A034388, A036507-A036536.
%Y A037069 Cf. A037053, A037055, A037057, A037059, A037061, A037063, A037065, A037067, A037071.
%K A037069 nonn,base,easy
%O A037069 0,1
%A A037069 _Patrick De Geest_, Jan 04 1999
%E A037069 Corrected by _Jud McCranie_, Jan 04 2001
%E A037069 More terms from _Erich Friedman_, Jun 03 2001
%E A037069 More terms and a(0) = 2 from _M. F. Hasler_, Feb 22 2016