Question

Is there another term in this sequence, and if so what is it?

2,3,4, 82000, ?

The pattern is each number is represented as only 1s and 0s in different bases, so

2 is 10 in base 2

3 is 11 in base 2 and 10 in base 3

4 is 100 in base 2, 11 in base 3, and 10 in base 4

82000 is 10100000001010000 in base 2, 11011111001 in base 3, 110001100 in base 4 and 10111000 in base 5.

These are the smallest possible numbers meeting this requirement. So the next number has to be only represented as 1s and 0s in base 2,3,4,5, and 6.

Answer 1

It kinda seems less complicated if you rephrase it as a number that is only expressed as a sum of powers...

\[2 = 2^1\]\[3=2^1+2^0 = 3^1\]\[4=2^2=3^1+3^0=4^1\]\[82000=2^{16}+2^{14}+2^6+2^4=3^{10}+3^9+3^7+3^6+3^5+3^4+3^3+3^0= \\ 4^8+4^7+4^3+4^2=5^7+5^5+5^4+5^3 \]

Kind interesting, since this sorta implies that in order to work, the powers on 2 and powers on 4 are really the same, so only even powers of 2 will work.