When d(m) = d(m+n) = n

[Leroy Quet]

--- Leroy Quet <q1qq2qqq3qqqq at yahoo.com> wrote:

> I went ahead and submitted the first 10 terms
> of
> this sequence, as calculated by Maximilian
> Hasler..
> I assume that the question as to whether EVERY
> positive integer n leads to a solution is an
> open
> problem.

I mean here: "whether every EVEN positive integer
n leads to a solution" for the d(m) = d(m+n) = n
problem;
which stated another way is "whether EVERY
positive integer n leads to a solution" for the
d(m) = d(m+2n) = 2n problem.

Sorry if I created any confusion.

Leroy Quet

> (Or did I miss something in the
> discussion?)
> I forgot to add the comment that there is no
> solution for any ODD positive integer n to
> d(m)=d(m+n) = n.
> 
> Maybe someone else should submit the
> d(m)=d(m+n)=2n sequence, since I myself did not
> come up with that variation.
> 
> When my sequence below finally appears on the
> EIS, I encourage anyone who wants to post
> extensions/comments to do so.
> 
> Thanks always,
> Leroy Quet
> 
> PS: It has been an unusually long time since I
> posted this sequence for me to not yet have
> received an automated reply.
> Are things at the on-line EIS working okay? (It
> IS Sunday, one of the days things tend to break
> down, since those who would fix any problems
> have
> taken the weekend off...)
> 
> %I A139416
> %S A139416 3, 6, 12, 70, 600281, 60, 1458, 264,
> 450, 266875
> %N A139416 a(n) = smallest positive integer
> such
> that d(a(n)) = d(a(n)+2n) = 2n, where d(m) is
> the
> number of positive divisors of m.
> %C A139416 Does this sequence have a term for
> every positive integer n, or are there no
> solutions for some n?
> 
> First 10 terms calculated by Maximilian Hasler.
> %e A139416 For a(4) we want the smallest
> integer
> m such that d(m) = d(m+8) = 8. The positive
> integers that have 8 divisors each form the
> sequence: 24, 30, 40, 42, 54, 56, 66, 70, 78,
> 88,
> 102, 104, 105, 110,...(A030626)
> The first (not necessarily adjacent) pair of
> integers with 8 divisors each that is separated
> by exactly 8 is (70,78). So a(4) is the least
> element of this pair, which is 70.
> %O A139416 1
> %K A139416 ,more,nonn,



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