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A020665 a(n) is the (conjectured) maximal exponent k such that n^k does not contain a digit zero in its decimal expansion.

%I A020665 #74 Feb 16 2025 08:32:34
%S A020665 86,68,43,58,44,35,27,34,0,41,26,14,34,27,19,27,17,44,0,13,22,10,13,
%T A020665 29,15,9,16,14,0,16,7,23,5,17,22,16,10,19,0,9,13,10,6,39,7,8,19,5,0,
%U A020665 19,18,7,13,11,23,7,23,14,0,16,5,14,12,3,14,14,14,12,0,8,22,6,4,19,11,12,10,9,0
%N A020665 a(n) is the (conjectured) maximal exponent k such that n^k does not contain a digit zero in its decimal expansion.
%C A020665 Most of these values are not proved rigorously, but the search has been pushed very large (~ 10^9 or beyond for many n). See the OEIS wiki page for further reading. - _M. F. Hasler_, Mar 08 2014
%C A020665 From _Bill McEachen_, Apr 01 2015: (Start)
%C A020665 It appears that the values at square pointers will be no more than that of the base pointer. Specifically when the value at the base pointer is even, the value at the square will be 50%. For example, the sequence n=2,4,16 yields a(n)=86,43,19. The sequence n=3,9,81 yields a(n)=68,34,17.
%C A020665 Values at other than squares are less obvious. However, at some point, the run of the squares ends, implying remaining nonzero values should indicate either nonsquares or prime entries. (End)
%C A020665 Since (n^b)^j = n^(b*j), a(n) >= b*a(n^b); if a(n) is divisible by b then a(n^b) = a(n)/b. - _Robert Israel_, Apr 01 2015
%H A020665 Antoine Beaulieu, <a href="/A020665/b020665.txt">Table of n, a(n) for n = 2..200</a> (checked with PARI up to k=10^5)
%H A020665 M. F. Hasler, <a href="https://oeis.org/wiki/Zeroless_powers">Zeroless powers</a>, OEIS Wiki, Mar 07 2014
%H A020665 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Zero.html">Zero</a>
%F A020665 a(10n) = 0 for any n>0. - _M. F. Hasler_, Dec 17 2014
%F A020665 a(100n+1) = 0 for any n>0. - _Robert Israel_, Apr 01 2015
%F A020665 a(80*n+65) <= 3, because for k >= 4, (80*n+65)^k == 625 (mod 10000). - _Robert Israel_, Apr 02 2015
%F A020665 From _Chai Wah Wu_, Jan 08 2020: (Start)
%F A020665 The following values and bounds are for the actual maximal exponents (not conjectured).
%F A020665 a(A052382(n)) > 0 for n > 1.
%F A020665 a(225) = 1
%F A020665 a(225^k) = 0 for k > 1.
%F A020665 a(625) = 1.
%F A020665 a(625^k) = 0 for k > 1.
%F A020665 a(3126) = 2.
%F A020665 a(3126^2) = 1.
%F A020665 a(3126^k) = 0 for k > 2.
%F A020665 a(9376) = 1.
%F A020665 a(9376^k) = 0 for k > 1.
%F A020665 a(21876) = 2.
%F A020665 a(21876^2) = 1.
%F A020665 a(21876^k) = 0 for k > 2.
%F A020665 a(34376) = 1.
%F A020665 a(34376^k) = 0 for k > 1.
%F A020665 a(400*n + 225) <= 1, since for k >= 2, (400*n + 225)^k == 625 (mod 10000), i.e., if 400*n + 225 is in A052382, then a(400*n+225) = 1, otherwise it is 0.
%F A020665 a(25000*n + 34376) <= 1, since for k >= 2, (25000*n + 34376)^k == 9376 (mod 100000), i.e., if 25000*n + 34376 is in A052382, then a(25000*n + 34376) = 1, otherwise it is 0.
%F A020665 a(25000*n + 21876) <= 2, since for k >= 3, (25000*n + 21876)^k == 9376 (mod 100000).
%F A020665 a(12500*n + 3126) <= 4, since for k >= 5, (12500*n + 3126)^k == 9376 (mod 100000).
%F A020665 (End)
%e A020665 a(13) = 14 because 13^14 does not have a digit 0, but (it is conjectured that) for all k > 14, 13^k will have a digit 0. It is not excluded that there may be some k < a(n) for which n^k does have a digit 0, as is the case for 13^6. - _M. F. Hasler_, Mar 29 2015
%p A020665 f:= proc(n)
%p A020665   local p;
%p A020665   if n mod 10 = 0 then return 0 fi;
%p A020665   for p from 100 by -1 do
%p A020665     if not has(convert(n^p,base,10),0) then return(p) fi
%p A020665   od
%p A020665 0
%p A020665 end proc:
%p A020665 seq(f(n),n=2..80); # _Robert Israel_, Apr 01 2015
%t A020665 a = {}; Do[ If[ Mod[n, 10] == 0, b = 0; Continue]; Do[ If[ Count[ IntegerDigits[n^k], 0 ] == 0, b = k], {k, 1, 200} ]; a = Append[a, b], {n, 2, 81} ];
%o A020665 (PARI) Nmax(x,L=99,m=0)=for(n=1,L,vecmin(digits(x^n))&&m=n);m \\ L=99 is enough to reproduce the known results, since no value > 86 is known; _M. F. Hasler_, Mar 08 2014
%Y A020665 For the zeroless numbers (powers x^n), see A238938, A238939, A238940, A195948, A238936, A195908, A195946, A195945, A195942, A195943, A103662.
%Y A020665 For the corresponding exponents, see A007377, A008839, A030700, A030701, A008839, A030702, A030703, A030704, A030705, A030706, A195944.
%Y A020665 For other related sequences, see A011540, A052382, A027870, A102483, A103663.
%K A020665 nonn,base
%O A020665 2,1
%A A020665 _David W. Wilson_