Positive integers of the form $(a+b+c)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})$

I started investigating which integer values can $f(a,b,c):=(a+b+c)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})$ take for $a,b,c$ being positive integers.

For all $a,b,c>0$ we have $f(a,b,c)\ge 9$.

For $a=b=c$ (any value) we have $f(a,a,a)=f(1,1,1)=9$.

Also $f(1,2,2)=10$ and $f(2,3,6)=11$.

I have no example for 12.

Is the set of integer values of $f$ finite or not?

I'm pretty sure that there are infinitely many natural numbers that cannot be values of $f$, but can't prove it.

• See the sequence of such integers in OEIS: oeis.org/A085514 – Steve Kass Sep 4 '16 at 22:42
• – Will Jagy Sep 4 '16 at 22:57
• @WillJagy Augh, my eyes! Good referene though. – 6005 Sep 4 '16 at 22:58
• $f(a,b,c)$ is an integer if and only if $abc \mid (a+b)(b+c)(c+a)$ – 6005 Sep 4 '16 at 22:59
• Clearly if a=b=c=n will work so might as well assume mutual gcd is 1. If b=c=1 a =2 is possible. – fleablood Sep 5 '16 at 0:28

$$a \leq b \leq c \leq 1000, \; \; \; \gcd(a,b,c) = 1$$

9    was ratio     1  1  1      LHS 9  RHS 1   ratio  9
10    was ratio     1  1  2      LHS 20  RHS 2   ratio  10
10    was ratio     1  2  2      LHS 40  RHS 4   ratio  10
11    was ratio     1  2  3      LHS 66  RHS 6   ratio  11
11    was ratio     2  3  6      LHS 396  RHS 36   ratio  11
14    was ratio     2  3  10      LHS 840  RHS 60   ratio  14
14    was ratio     3  10  15      LHS 6300  RHS 450   ratio  14
15    was ratio     1  2  6      LHS 180  RHS 12   ratio  15
15    was ratio     1  3  6      LHS 270  RHS 18   ratio  15
18    was ratio     2  10  15      LHS 5400  RHS 300   ratio  18
18    was ratio     2  3  15      LHS 1620  RHS 90   ratio  18
26    was ratio     1  6  14      LHS 2184  RHS 84   ratio  26
26    was ratio     3  7  42      LHS 22932  RHS 882   ratio  26
30    was ratio     10  77  165      LHS 3811500  RHS 127050   ratio  30
30    was ratio     14  30  231      LHS 2910600  RHS 97020   ratio  30
34    was ratio     1  6  21      LHS 4284  RHS 126   ratio  34
34    was ratio     2  7  42      LHS 19992  RHS 588   ratio  34
35    was ratio     3  10  65      LHS 68250  RHS 1950   ratio  35
35    was ratio     6  39  130      LHS 1064700  RHS 30420   ratio  35
55    was ratio     1  14  35      LHS 26950  RHS 490   ratio  55
55    was ratio     2  5  70      LHS 38500  RHS 700   ratio  55
59    was ratio     2  15  85      LHS 150450  RHS 2550   ratio  59
59    was ratio     6  34  255      LHS 3069180  RHS 52020   ratio  59
63    was ratio     5  119  170      LHS 6372450  RHS 101150   ratio  63
63    was ratio     7  10  238      LHS 1049580  RHS 16660   ratio  63
74    was ratio     11  135  594      LHS 65274660  RHS 882090   ratio  74
74    was ratio     5  22  270      LHS 2197800  RHS 29700   ratio  74
105    was ratio     1  21  77      LHS 169785  RHS 1617   ratio  105
105    was ratio     3  11  231      LHS 800415  RHS 7623   ratio  105
126    was ratio     3  114  247      LHS 10643724  RHS 84474   ratio  126
126    was ratio     6  13  494      LHS 4855032  RHS 38532   ratio  126
131    was ratio     1  35  90      LHS 412650  RHS 3150   ratio  131
131    was ratio     7  18  630      LHS 10398780  RHS 79380   ratio  131
179    was ratio     3  65  442      LHS 15428010  RHS 86190   ratio  179
190    was ratio     2  21  322      LHS 2569560  RHS 13524   ratio  190
190    was ratio     3  46  483      LHS 12664260  RHS 66654   ratio  190
294    was ratio     5  570  874      LHS 732324600  RHS 2490900   ratio  294
297    was ratio     5  513  945      LHS 719905725  RHS 2423925   ratio  297
298    was ratio     2  85  493      LHS 24975380  RHS 83810   ratio  298
319    was ratio     2  150  475      LHS 45457500  RHS 142500   ratio  319
323    was ratio     2  75  550      LHS 26647500  RHS 82500   ratio  323
323    was ratio     3  22  825      LHS 17587350  RHS 54450   ratio  323
326    was ratio     2  55  570      LHS 20440200  RHS 62700   ratio  326
330    was ratio     1  90  234      LHS 6949800  RHS 21060   ratio  330
370    was ratio     1  77  286      LHS 8148140  RHS 22022   ratio  370
851    was ratio     1  234  611      LHS 121670874  RHS 142974   ratio  851