Quote:
Originally Posted by Dr Sardonicus
No. The first six highly composite numbers are 1, 2, 4, 6, 12, and 24.
We have b_{5} = lcm([3,5,17,65,4097]) = 3*5*13*17*241, and
2^{24} + 1 = 16777217 = 97*257*673. Thus
c_{6} = (2^{24} + 1)/gcd(3*5*13*17*241,97*257*673) = (2^{24} + 1)/1 = 97*257*673

Thanks Dr Sardonicus.
I didn't think a counter example could be found for such small
. My mistake was in assuming that the nonprimitive part of
must contain many factors from smaller
where
is another highly composite integer. However just because
is highly composite, doesn't mean that
.
In this case, even considering only the primitive part of
which is
we still get a composite integer, which is kind of surprising because we can also prove that any prime factor of the primitive part of
must be of the form
, which greatly restricts the possible number of "eligible" prime factors to nearly 1% of all primes that the are in the range from
. However when
, we get
which is a factor lol!