# All about a failed conjecture.

Some months ago I made the following conjecture -
Let $d(n)$ denote the number of divisors of $n$.
Then let $N$ be a number such that $d(N)$ divides $N$ . Also let $I= \frac{N}{d(N)}$ which is defined as the "Index of Beauty of $N$ ". Then, For every number $I$ there exists a number $N$ such that $I$ is the index of beauty of $N$.

This conjecture was proved false by Greg Martin here.
He said that it can be showed by exaustive computation that the following $I$ fail the conjecture under $1000$ are $\{18, 27, 30, 45, 63, 64, 72, 99, 105, 112, 117, 144, 153, 160, 162, 165, 171, 195, 207, 225, 243, 252, 255, 261, 279, 285, 288, 294, 320, 333, 336, 345, 352, 360, 369, 387, 396, 405, 416, 423, 435, 441, 465, 468, 477, 490, 504, 531, 544, 549, 555, 567, 576, 603, 608, 612, 615, 616, 625, 639, 645, 657, 684, 705, 711, 726, 728, 735, 736, 747, 792, 795, 801, 810, 828, 840, 873, 880, 885, 891, 909, 915, 927, 928, 936, 952, 960, 963, 981, 992\}$

Now what I am interested is that sequence of $I$ that fails.
(i) Is this sequence infinite?How?
(ii)Is there any approximation which can tell the number of such failed $I$ less than a fixed $x$
(iii)If the sequence is infinite then are there canonical forms in which all of the values are in our list.

• – Robert Israel Feb 18 '14 at 7:54
• @RobertIsrael this did not answer any of my 3 question.. – Shivam Patel Feb 18 '14 at 8:26
• I didn't say it did. It's just to show that the problem has been looked at before, and more values computed. – Robert Israel Feb 18 '14 at 16:22
• @RobertIsrael can you tell me what are the best known upper and lower bonds on $\frac{n}{d(n)}$? – Shivam Patel Feb 19 '14 at 7:05
• Upper is easy: $n/d(n) = n/2$ if $n$ is prime. An easy lower bound is $n/d(n) \ge \sqrt{n}/2$, but I would expect there to be better lower bounds. – Robert Israel Feb 19 '14 at 7:54