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

[Dean Hickerson]

"Max A." <maxale at gmail.com> wrote:

> Note that d(m) <= log2(m), implying that m/d(m) >= m/log2(m) tends to
> infinity as m grows.

This is false for m=1, 2, 3, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, ...

> 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.

Actually, all of those occur as values of floor(m/d(m)), for these
values of m:

42286560, 34355097, 30124176, 36425484, 50762624, 38646615, 38726020, 34916880,
31507452, 31823235, 31897188, 32691420, 43734888, 32846004, 43852180, 32982543,
44221828, 33180615, 33715900, 37531344, 30040448, 30043048, 30051320, 30102968,
30135880, 30262120, 34134508, 30371080, 30403944, 30443720, 30466605, 30627320,
38362624, 30711640, 30771309, 30803072, 30807424, 30941715, 34883388, 46521525,
31019247, 31152256, 31158656, 31292680, 31329848, 47087667, 31469696, 31471448,
39370640, 31537688, 31643227, 47531575, 31729352, 31740544, 31773681, 31799464,
31895375, 31901877, 63944769

I don't know the answer to Leroy's first question.  The answer to the second
one (Is the number of m's where floor(m/d(m)) = n finite for all n?) is "yes",
since  d(m) < 2 sqrt(m)  for all m.  (Every divisor of m is either <= sqrt(m)
or has the form m/k where k is a divisor <= sqrt(m).)  So  m/d(m) > sqrt(m)/2,
which tends to infinity with m.  (Much better upper bounds on d(m) are known,
but I don't have a reference handy at the moment.)

Dean Hickerson
dean at math.ucdavis.edu