Intended for: July 21, 2012
Timetable
- First draft entered by Alonso del Arte on April 29, 2011 based on a verbatim copy of a write-up from November 13, 2010. ✓
- Draft reviewed by Alonso del Arte on July 16, 2011 ✓
- Draft approved by Daniel Forgues on July 20, 2011 ✓
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A175607 Largest number
such that the
greatest prime factor of
is
.
-
{ 3, 17, 161, 8749, 19601, 246401, 672281, ... }
It surprised me, but after a little reflection, it makes a lot of sense: for any prime
, there is a short (that is to say, finite) list of numbers
such that
has
as its greatest prime factor. (See:
Greatest prime factor of n 2 − 1, n ≥ 2.) For example, for
, we find that
3 2 − 1 = 8 = 2 3. No larger number satisfies
.
Even-indexed powers of two (
A000302) are squares of a smaller power of two, and therefore
is not an integer if
is even. The reason why higher
odd-indexed powers of two (
A004171) don’t work for this purpose is, as so many math textbooks say, left as an exercise for the Reader.
According to Artur Jasinski, every prime
has a corresponding
with the property described here.