friendly/solitary numbers [was: typos]

[Dean Hickerson]

Eric Weisstein asked:

> It it known if 10, 14, 15, 20, 22, etc. are actually friendly or solitary?  

No.

There are some numbers which can easily be proved to be solitary.  This
includes all numbers n for which n and sigma(n) are relatively prime, and
some others, like:

    18, 45, 48, 52, 136, 148, 160, 162, 176, 192, 196, 208, 232, 244,
    261, 272, 292, 296, 297, 304, 320, 352, 369

Some numbers can be proved to be friendly by finding another integer with the
same index.  Sometimes the smallest such number is fairly large; e.g. 24 is
friendly because index(24) = index(91963648).

But there are many numbers whose status is unknown (at least by me),
including:

    10, 14, 15, 20, 22, 26, 33, 34, 38, 44, 46, 51, 54, 58, 62, 68, 69,
    70, 72, 74, 76, 82, 86, 87, 88, 90, 91, 92, 94, 95, 99, 104, 105, 106

> Is there a more recent reference than 
>
> Anderson, C. W and Hickerson, D. Problem 6020. "Friendly Integers." 
> Amer. Math. Monthly 84, 65-66, 1977.

Not that I know of.  In 1996 Carl Pomerance told me that he could prove
that the solitary numbers have positive density, disproving a conjecture that
Anderson and I had made.  (It's easy to see that the friendly numbers have
positive density.)  But I haven't seen his proof, and I don't know if he
published it.

Dean Hickerson
dean at math.ucdavis.edu