d(m) = d(m+1) = 2n
Leroy Quet
q1qq2qqq3qqqq at yahoo.com
Fri Aug 15 02:47:03 CEST 2008
Never mind. It seems quite obvious that there are no more m's or n's.
Someone please correct me if I am wrong:
Let us say that m has 2n divisors. Then there are a number of ways the exponents in the prime-factorization of m, independent of the primes in the prime-factorization, can be written. (Since 2n = product(1 + e_k), where each e_k is an exponent in the prime-factorization.)
Since m is the SMALLEST m with 2n divisors, then the primes dividing m are all consecutive primes starting with 2.
But m+1 is coprime to m. So its primes must differ from those dividing m, and therefore be larger than the largest prime dividing m.
Now, it is possible that the exponents in the prime-factorization of m+1 (which lead to the number of divisors of m+1 being 2n) would allow m+1 to be very close to m (one away from m, in fact), because the prime-factorization exponents of m+1 are smaller than those of m. But if that is so, then replacing the primes that divide m+1 with consecutive primes starting at 2, using the same prime-factorization exponents, would produce an integer <= m.
This is all unrigorous, of course. But I am tired and lazy. The makings of the start of a rigorous proof are here, hopefully.
Thanks,
Leroy Quet
--- On Thu, 8/14/08, Leroy Quet <q1qq2qqq3qqqq at yahoo.com> wrote:
> From: Leroy Quet <q1qq2qqq3qqqq at yahoo.com>
> Subject: d(m) = d(m+1) = 2n
> To: seqfan at ext.jussieu.fr
> Cc: qq-quet at mindspring.com
> Date: Thursday, August 14, 2008, 9:38 PM
> Consider the two sequences, one of positive integers m and
> one of positive integers n:
> m and n are such that m is the smallest positive integer
> with 2n divisors, and m+1 also has 2n divisors.
>
> For example, 2 has 2 divisors, and 2+1 also has 2 divisors.
> And 2 is the smallest positive integer with 2 divisors.
>
> So the m sequence starts: 2,...
> And the n sequence starts: 1,...
>
> This may be easily answered, I bet, but are there any other
> m's and n's?
>
> Thanks,
> Leroy Quet
Alexander, thanks for catching this error - what happened
was that I meant to delete A115400, but accidentally deleted A115401
instead.
There will be a new version of the OEIS in a couple of
hours with this corrected
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