A263450 - OEIS

%I #21 Oct 26 2015 22:43:10

%S 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,

%T 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,

%U 6,2,2,2,4,3,5,4,6,4,2,1,1,3,4,3,5,3,3,2,2,5

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

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

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

%C From _Robert Israel_, Oct 19 2015: (Start)

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

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

%H Robert Israel, <a href="/A263450/b263450.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A071531(prime(n)). - _Michel Marcus_, Oct 21 2015

%p f:= proc(m) local k;

%p for k from 1 do

%p      if has(convert(m^k,base,10),0) then return k fi

%p    od

%p end proc:

%p seq(f(ithprime(i)), i=1..1000); # _Robert Israel_, Oct 19 2015

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

%o (PARI) a(n) = {p = prime(n); k = 1; while (vecmin(digits(p^k)), k++); k;} \\ _Michel Marcus_, Oct 21 2015

%Y Cf. A062584, A071531, A103662, A103663.

%K nonn,base

%O 1,1

%A _Zak Seidov_, Oct 18 2015