login
Primes p such that p^k is zeroless for k=1,...,6.
5

%I #19 Sep 07 2018 17:42:29

%S 2,3,5,17,48989,5453971,61636943111479,128359315177123,

%T 884785266899689,1116777231836989

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

%C Primes in A253647; both sequences are conjectured to be finite.

%C 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 The terms 2, 3 and 5 seem to be the only primes in A124648, i.e., satisfy the restriction up to k=7.

%C a(7) > 10^11. - _Chai Wah Wu_, Jan 10 2015

%C a(11) > 3.3*10^16. - _Giovanni Resta_, Sep 06 2018

%t Select[Prime[Range[10^7]],Count[Flatten[IntegerDigits/@(#^Range[6])],0] == 0&] (* _Harvey P. Dale_, May 26 2016 *)

%o (PARI) forprime(p=0,,forstep(k=6,1,-1,vecmin(digits(p^k))||next(2));print1(p","))

%o (Python)

%o from sympy import isprime

%o A253646_list = [2]

%o for i in range(1,10**6,2):

%o ....if not '0' in str(i):

%o ........m = i

%o ........for k in range(5):

%o ............m *= i

%o ............if '0' in str(m):

%o ................break

%o ........else:

%o ............if isprime(i):

%o ................A253646_list.append(i) # _Chai Wah Wu_, Jan 10 2015

%Y Cf. A253647, A253645, A124648.

%K nonn,base,hard,more

%O 1,1

%A _Zak Seidov_ and _M. F. Hasler_, Jan 07 2015

%E a(7)-a(10) from _Giovanni Resta_, Sep 03 2018