Re: smallest positive integer that has exactly k divisors

Mike Amling <dr-ahmadinejad@xxxxxxxxxxxx> writes:
Phil Carmody wrote:
"mensanator@xxxxxxxxxxx" <mensanator@xxxxxxx> writes:
## Did you really mean 2000 factors? You do know that the
## number of divisors is 2**factors, don't you?
Really? Then how can 12 have 6 divisors?
It doesn't, it has 6 UNIQUE divisors.

I haven't been following this thread, so if this answer has been
given before, please forgive the redundancy.
Assume the OP is interested in the number of unique divisors of
n. That is the product of the exponents, each incremented by 1, in the
prime factorization of n (counting 1 and n as divisors). So, given k,
the desired number of factors of n, first factor k into primes. Say
there are x (not necessarily unique) prime factors of k. The desired n
is the product of the smallest x prime numbers, in increasing order,
raised to exponents which are one less than the prime factors of k, in
decreasing order.
So, what is the smallest integer with 18 divisors? k=18=3*3*2, so
n=(2**(3-1))*(3**(3-1))*(5**(2-1))=180, with divisors 1, 2, 3, 4, 5,
6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180.

That technique fails for k as small as 8.
You can sometimes stick a bunch of extra 2s in more cheaply than
adding an extra prime at the end.

Phil
--
Dear aunt, let's set so double the killer delete select all.
-- Microsoft voice recognition live demonstration
.

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