A263450 Smallest integer k>0 such that there is at least one zero in the decimal representation of prime(n)^k.

10, 10, 8, 4, 5, 6, 7, 4, 6, 4, 6, 3, 5, 3, 2, 2, 3, 5, 3, 2, 3, 3, 5, 3, 2, 1, 1, 1, 1, 4, 3, 3, 6, 4, 2, 2, 4, 3, 5, 4, 2, 4, 4, 3, 2, 2, 5, 3, 3, 3, 6, 4, 2, 2, 2, 4, 3, 3, 5, 3, 2, 4, 1, 3, 3, 2, 2, 6, 2, 2, 2, 4, 3, 5, 4, 6, 4, 2, 1, 1, 3, 4, 3, 5, 3, 3, 2, 2, 5

Offset: 1

Author

Zak Seidov, Oct 18 2015

Keywords

• nonn
• base

Comments

Conjecture: there are an infinite number of ones in the sequence.

Corresponding values of prime(n)^k: 1024, 59049, 390625, 2401, 161051, 4826809, 410338673, 130321 (not yet in OEIS).

From Robert Israel, Oct 19 2015: (Start)

By Dirichlet's theorem there are infinitely many n for which prime(n) == 1 (mod 100), and these all have a(n) = 1.

All a(n) <= 20, since every x coprime to 10 has x^20 == 1 (mod 100). (End)

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A062584, A071531, A103662, A103663.

Programs

Maple

f:= proc(m) local k;
for k from 1 do
     if has(convert(m^k,base,10),0) then return k fi
   od
end proc:
seq(f(ithprime(i)), i=1..1000); # Robert Israel, Oct 19 2015

Mathematica

Reap[Do[p=Prime[n];k=1;While[Min[IntegerDigits[p^k]]>0,k++];Sow[k],{n,1,200}]][[2,1]]

PARI

a(n) = {p = prime(n); k = 1; while (vecmin(digits(p^k)), k++); k;} \\ Michel Marcus, Oct 21 2015

Formula

a(n) = A071531(prime(n)). - Michel Marcus, Oct 21 2015