%I #18 Sep 09 2024 09:34:57
%S 1,3,22557,354081,419163,286623930087
%N a(n) is the least k such that the concatenations of k^i, i=0..j, are prime for j from 1 to n.
%C For n >= 2, a(n) == 3 (mod 6) if it exists.
%C a(6) > 3 * 10^9 if it exists.
%C No further terms < 10^11, testing all odd numbers (see Python program). - _Michael S. Branicky_, Sep 08 2024
%e a(3) = 22557: 22557^2 = 508818249, 22557^3 = 11477413242693, and the concatenations 122557, 122557508818249, and 12255750881824911477413242693 are all prime.
%p tcat:= (a,b) -> a*10^(1+ilog10(b))+b:
%p k:= 2: R:= 1:
%p for n from 2 to 5 do
%p for j from k by 6 do
%p c:= 1; good:= true;
%p for i from 1 to n while good do
%p c:= tcat(c, j^i);
%p if not isprime(c) then good:= false; break fi
%p od;
%p if good then k:= j; R:= R,j; break fi
%p od od:
%p R;
%o (Python)
%o from itertools import count
%o from gmpy2 import is_prime, mpz, digits
%o def f(k):
%o i, s = 0, "1"
%o while True:
%o ski = digits(k**(i+1))
%o if not is_prime(mpz(s+ski)):
%o break
%o i, s = i+1, s+ski
%o return i
%o def afind():
%o adict = dict()
%o for m in count(1, 2): # or count(3, 6) for a(2) on per comment
%o v = f(m)
%o if v > 0 and v not in adict:
%o adict[v] = m
%o print("FOUND", m, v)
%o afind() # _Michael S. Branicky_, Sep 08 2024
%Y Cf. A096469.
%K nonn,base,more
%O 1,2
%A _Robert Israel_, Sep 06 2024
%E a(6) from _Michael S. Branicky_, Sep 09 2024