First Occurrence Of Floor(m/d(m))

[Max A.]

Note that d(m) <= log2(m), implying that m/d(m) >= m/log2(m) tends to
infinity as m grows.
Therefore, for a fixed n there is only a finite number of solutions to
floor(m/d(m))=n, implying that there indeed exists a LARGEST solution
(answer to question 2).

For the following n below 10^6 there is NO solution to floor(m/d(m))=n:

440485, 715731, 753104, 758864, 793166, 805137, 806792, 872922,
875207, 883978, 886033, 908095, 911143, 912389, 913587, 916181,
921288, 921683, 936552, 938283, 938764, 938845, 939103, 940717,
941746, 945691, 948180, 949096, 950123, 951366, 952081, 957103,
959065, 959738, 961603, 962596, 962732, 966928, 968983, 969198,
969351, 973508, 973708, 977896, 979057, 980993, 983428, 983482,
984266, 985552, 988850, 990241, 991542, 991892, 992927, 993733,
996730, 996933, 999137

To verify that it is enough to check all m below 3*10^7 since
3*10^7/log2(3*10^7) > 10^6.

Max

On 12/30/06, Leroy Quet <qq-quet at mindspring.com> wrote:
> I just submitted this sequence:
>
> >%I A126888
> >%S A126888 1,5,7,28,11,13,44,17,19,63,23,51,55,29
> >%N A126888 a(n) is the smallest positive integer such that
> >floor(a(n)/d(a(n))) = n, where d(m) is the number of positive divisors of m.
> >%C A126888 Does every term have a value? ie, Does every positive integer n
> >equal floor(m/d(m)) for some m?
> >%Y A126888 A126889,A078709
> >%O A126888 1
> >%K A126888 ,more,nonn,
>
> (Hopefully I didn't errr...)
>
> Questions:
>
> 1) As noted in the C-line, I don't know if every term has a value. Do
> they? (I guess if some n's don't equal any floor(m/d(m))'s, then those
> terms of the sequence can just be 0, an arbitrary fix.)
>
> 2) Is there a related sequence of LARGEST positive integers {a(n)} where
> floor(a(n)/d(a(n))) = n?
> In other words, is the number of m's where floor(m/d(m)) = n finite for
> all n? For some n?
>
> (I bet this can all be answered by Hardy and Wright, but I don't have a
> copy.)
>
> Thanks,
> Leroy Quet
>