OFFSET
1,1
COMMENTS
It is unknown if a(n) ends in a 9 for all n > 5.
a(n) <= 5*2^n-1 because (5*2^n-1)^(5^(n-1)) = -1 (mod 10^n), (5*2^n-1)^k != -1 (mod 10^(n+1)), and (5*2^n-1)^k != 11 (mod 100). - Hiroaki Yamanouchi, Jul 11 2014
FORMULA
Conjecture for n>5: a(n) = 10*2^(n-1)-1, k = (2r+1)*5^(n-2), r>=0. - Lars Blomberg, Jul 08 2014
EXAMPLE
33^k ends in 4 identical digits (let k == 187 mod 500) and never ends in more than 4 identical digits. Since this is not true for 2 <= m <= 32, a(4) = 33.
PROG
(PARI) a(n, p)=for(c=0, 10^p, st=Str(n^c); if(#st>p, jo=(eval(st)%(10^p)); if(jo==1, return(c)); if(ispower(jo, , &k), if(ispower(n)!=0, if(jo^(1/(ispower(jo, , &k)))==(n^(1/(ispower(n)))), return(c))); if(ispower(n)==0, if(jo==n^(ispower(jo, , &k)), return(c))))))
hup(x)={m=1; for(i=2, 100, f=0; for(j=m, a(x, i), dt=(x^j)%(10^i); b=""; for(w=1, i, b=concat(b, "1")); if(dt%eval(b)==0, f++; r=j; break)); if(f==0, return(i-1)); m=r)}
n=1; while(n<100, for(x=2, 10^3, if(hup(x)==n, print1(x, ", "); break)); n++)
CROSSREFS
KEYWORD
nonn,base,hard,more
AUTHOR
Derek Orr, Jun 26 2014
EXTENSIONS
a(9)-a(27) from Lars Blomberg, Jul 08 2014
Definition and example edited by Robert Israel, Jul 10 2014
STATUS
approved