A385332 Integers k such that the set {k, k^2, ..., k^9} contains at least 8 zeroless numbers.

1, 2, 3, 5, 6, 11, 17, 68, 121, 786

Offset: 1

Author

Gonzalo Martínez, Jun 25 2025

Keywords

• nonn
• base
• more

Comments

1, 2, 3, 6 and 68 are the only integers less than 3.3*10^16 such that {k, k^2,..., k^8} are all zeroless (See A124649), but of them 1^9, 2^9 and 3^9 are the only zeroless ones. If we weaken the condition and ask that 8 of the 9 numbers in the set {k, k^2,..., k^9} are zeroless, then more numbers appear that satisfy this property.

Is a(10) = 786 the largest term?

Examples

			5 is a term, since the first nine powers of 5 are: 5, 25, 125, 625, 3125, 15625, 78125, 390625 and 1953125, where 8 of the 9 are zeroless.
		

Crossrefs

Cf. A124648, A124649.

Programs

Maple

zless:= n -> not has(convert(n,base,10),0):
filter:= proc(n) local i; nops(select(zless, [seq(n^i,i=1..9)]))>=8 end proc:
select(filter, [$1..1000]); # Robert Israel, Jun 26 2025

Mathematica

Select[Range[10^5],Total[Boole/@Positive/@Min/@IntegerDigits/@(#^Range[9])]>7&] (* James C. McMahon, Jul 01 2025 *)