

A020665


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


39



86, 68, 43, 58, 44, 35, 27, 34, 0, 41, 26, 14, 34, 27, 19, 27, 17, 44, 0, 13, 22, 10, 13, 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, 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
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OFFSET

2,1


COMMENTS

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
From Bill McEachen, Apr 01 2015: (Start) 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.
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)
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


LINKS

Table of n, a(n) for n=2..80.
M. F. Hasler, Zeroless powers, OEIS Wiki, Mar 07 2014
Eric Weisstein's World of Mathematics, Zero


FORMULA

a(10n) = 0 for any n>0.  M. F. Hasler, Dec 17 2014
a(100n+1) = 0 for any n>0.  Robert Israel, Apr 01 2015
a(80*n+65) <= 3, because for k >= 4, (80*n+65)^k == 625 mod 10000.  Robert Israel, Apr 02 2015


EXAMPLE

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 it is the case for 13^6.  M. F. Hasler, Mar 29 2015


MAPLE

f:= proc(n)
local p;
if n mod 10 = 0 then return 0 fi;
for p from 100 by 1 do
if not has(convert(n^p, base, 10), 0) then return(p) fi
od
0
end proc:
seq(f(n), n=2..80); # Robert Israel, Apr 01 2015


MATHEMATICA

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} ];


PROG

(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


CROSSREFS

For the zeroless numbers (powers x^n), see A238938, A238939, A238940, A195948, A238936, A195908, A195946, A195945, A195942, A195943, A103662.
For the corresponding exponents, see A007377, A008839, A030700, A030701, A008839, A030702, A030703, A030704, A030705, A030706, A195944.
For other related sequences, see A052382, A027870, A102483, A103663.
Sequence in context: A045916 A033406 A143759 * A259084 A058907 A045101
Adjacent sequences: A020662 A020663 A020664 * A020666 A020667 A020668


KEYWORD

nonn,base


AUTHOR

David W. Wilson


STATUS

approved



