In this short set of numbers two do not belong:
1, 16, 64, 512, 4096, 46656, 1000000, 2985984

Why?

The numbers which belong have two special properties. Those not conforming have one of these which is not the same for each of them.

Of the numbers – 1, 16, 64, 512, 4096, 46656, 1000000, and 2985984 – there are several different properties that can isolate a pair.

• 4096 and 2985984 are the only two of the eight numbers that cannot be octal (base-8) numbers. [Nor can they be septenary (base-7) or nonary (base-9) numbers].
• 64 and 512 have an even sum of their decimal digits and each of the other numbers have an odd sum of their decimal digits.
• 46656 and 2985984 are evil numbers and the other numbers are odious numbers. (A nonnegative integer in its binary expansion with an even number of 1s is termed evil and an odd number of 1s is termed odious.)

But these isolations are not where two properties are shared with the seven of the eight numbers with the other two numbers sharing each only one. For numbers of this restriction, (already noted in Kenny M's post ) there is the following:

• 16 and 512. 16 is the only number of the short set that does not result in an integer when the cube-root is taken; and 512 is the only number of the short set that does not result in an integer when the square-root is taken.

Posted by Dej Mar on 2007-04-30 02:31:11