The greatest length of a block of consecutive strange numbers is? A integer(>$0$)is said to be strange if all powers are odd in prime factorisation.

Question

A positive integer is said to be strange if all powers are odd in their prime factorisation. For instance, $22,23,24$ form a block of consecutive strange numbers because $22=2^1\times11^1,23=23^1,24=2^3\times3^1$. What is the greatest length of a block of consecutive strange numbers?

I tried all the numbers till 50 and obtained two series with $7$ terms which is the answer. The consecutive numbers are:- $$S_1=29,30,31,32,33,34,35\\S_2=37,38,39,40,41,42,43$$How can we make sure that we cannot find any series with more than $7$ terms?

Answer 1

If x is 4 mod 8, then x is not strange. Therefore, you can’t have a sequence of 8 consecutive strange numbers.