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

2, 3, 5, 17, 48989, 5453971, 61636943111479, 128359315177123, 884785266899689, 1116777231836989

Offset: 1

Author

Zak Seidov and M. F. Hasler, Jan 07 2015

Keywords

• nonn
• base
• hard
• more

Comments

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

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.

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

a(7) > 10^11. - Chai Wah Wu, Jan 10 2015

a(11) > 3.3*10^16. - Giovanni Resta, Sep 06 2018

Crossrefs

Cf. A253647, A253645, A124648.

Programs

Mathematica

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

PARI

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

Python

from sympy import isprime
A253646_list = [2]
for i in range(1,10**6,2):
    if not '0' in str(i):
        m = i
        for k in range(5):
            m *= i
            if '0' in str(m):
                break
        else:
            if isprime(i):
                A253646_list.append(i) # Chai Wah Wu, Jan 10 2015

Extensions

a(7)-a(10) from Giovanni Resta, Sep 03 2018