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A253646 Primes p such that p^k is zeroless for k=1,...,6.

%I A253646 #21 May 22 2025 10:21:41
%S A253646 2,3,5,17,48989,5453971,61636943111479,128359315177123,
%T A253646 884785266899689,1116777231836989
%N A253646 Primes p such that p^k is zeroless for k=1,...,6.
%C A253646 Primes in A253647; both sequences are conjectured to be finite.
%C A253646 The motivation for this sequence lies in the fact that many small primes satisfy the restriction up to k=5 (there are 52 terms below 10^6, cf. A253645), but including k=6 makes the sequence much sparser, with only one term between 17 and 5*10^6, and only one more term below 2*10^9.
%C A253646 The terms 2, 3 and 5 seem to be the only primes in A124648, i.e., satisfy the restriction up to k=7.
%C A253646 a(7) > 10^11. - _Chai Wah Wu_, Jan 10 2015
%C A253646 a(11) > 3.3*10^16. - _Giovanni Resta_, Sep 06 2018
%t A253646 Select[Prime[Range[10^7]],Count[Flatten[IntegerDigits/@(#^Range[6])],0] == 0&] (* _Harvey P. Dale_, May 26 2016 *)
%o A253646 (PARI) forprime(p=0,,forstep(k=6,1,-1,vecmin(digits(p^k))||next(2));print1(p","))
%o A253646 (Python)
%o A253646 from sympy import isprime
%o A253646 A253646_list = [2]
%o A253646 for i in range(1,10**6,2):
%o A253646     if not '0' in str(i):
%o A253646         m = i
%o A253646         for k in range(5):
%o A253646             m *= i
%o A253646             if '0' in str(m):
%o A253646                 break
%o A253646         else:
%o A253646             if isprime(i):
%o A253646                 A253646_list.append(i) # _Chai Wah Wu_, Jan 10 2015
%Y A253646 Cf. A253647, A253645, A124648.
%K A253646 nonn,base,hard,more
%O A253646 1,1
%A A253646 _Zak Seidov_ and _M. F. Hasler_, Jan 07 2015
%E A253646 a(7)-a(10) from _Giovanni Resta_, Sep 03 2018