What are all possible positive integers $k$ such that $k=\frac{a^2+b^2+c^2}{bc+ca+ab}$ for some positive integers $a$, $b$, and $c$?

There is such a solution if and only if both $k-1$ and $k+2$ have (well, different) integer expressions as some $u^2 + 3 v^2.$

The justification for that is in several answers I posted at

Find a solution: $3(x^2+y^2+z^2)=10(xy+yz+zx)$

$$ $$ $$ $$

Given $$ p^2 + 3 q^2 = 2 + k, $$ $$ r^2 + 3 s^2 = 4(k-1), $$ we can solve $$ (x^2 + y^2 + z^2) = k (yz + zx + xy) $$ with $$ x = 2 p^2 + 6 q^2 - p r - 3 p s + 3 q r - 3 q s, $$ $$ y = 2 p^2 + 6 q^2 - p r + 3 p s - 3 q r - 3 q s, $$ $$ z = 2 p^2 + 6 q^2 + 2 p r + 6 q s. $$

I did not immediately realize, the process of Vieta Jumping lets us take a mixed solution and create one with all the same $\pm$ sign. Suppose $x < 0,$ $y > 0,$ $z>0.$ We do a single jump: $$ x \mapsto k(y+z) - x, $$ where the new $x$ value is then positive!

The permissible values of your $k$ from 2 to 1000 are

  2      5     10     14     17     26     29     37     50     62
 65     74     77     82     98    101    109    110    122    125
145    149    170    173    190    194    197    209    226    242
245    257    269    290    302    305    314    325    334    362
365    398    401    410    434    437    442    469    482    485
497    509    514    530    554    557    577    590    602    605
626    629    674    677    685    689    701    722    725    730
770    773    785    794    830    842    845    869    874    890
901    917    962    965    973    974    989

These all lead to solutions $(a,b,c) $ where it may be that some variables are negative, some positive.

Let me work up some of the smallest such $k,$ see whether positive solutions appear.

$$ k = 17; \; \; \; (377,17,5) $$

$$ k = 26; \; \; \; (418,13,3) $$

$$ k = 29; \; \; \; (1109,11,27) $$

BY RECIPE .........................................

Mon Jul  6 19:11:55 PDT 2020

      2  ( 1, 1 , 4 )  p 1 q 1 r 1 s 1
      5  ( -1, 5 , 17 )   ( 111, 5 , 17 )  p 2 q 1 r 2 s 2
     10  ( 2, -1 , 5 )   ( 2, 71 , 5 )  p 0 q 2 r 3 s 3
     14  ( -1, 2 , 11 )   ( 183, 2 , 11 )  p 2 q 2 r 2 s 4
     17  ( -13, 23 , 47 )   ( 1203, 23 , 47 )  p 4 q 1 r 4 s 4
     26  ( 3, -2 , 13 )   ( 3, 418 , 13 )  p 1 q 3 r 5 s 5
     29  ( -7, 11 , 89 )   ( 2907, 11 , 89 )  p 2 q 3 r 2 s 6
     37  ( -11, 19 , 31 )   ( 1861, 19 , 31 )  p 6 q 1 r 6 s 6
     50  ( -5, 7 , 76 )   ( 4155, 7 , 76 )  p 2 q 4 r 2 s 8
     62  ( -5, 7 , 22 )   ( 1803, 7 , 22 )  p 4 q 4 r 1 s 9
     65  ( -61, 107 , 155 )   ( 17091, 107 , 155 )  p 8 q 1 r 8 s 8
     74  ( 22, -17 , 109 )   ( 22, 9711 , 109 )  p 1 q 5 r 7 s 9
     77  ( -13, 17 , 233 )   ( 19263, 17 , 233 )  p 2 q 5 r 2 s 10
     82  ( 5, -4 , 41 )   ( 5, 3776 , 41 )  p 3 q 5 r 9 s 9
     98  ( -4, 5 , 29 )   ( 3336, 5 , 29 )  p 5 q 5 r 5 s 11
    101  ( -97, 173 , 233 )   ( 41103, 173 , 233 )  p 10 q 1 r 10 s 10
    109  ( -29, 43 , 97 )   ( 15289, 43 , 97 )  p 6 q 5 r 0 s 12
    110  ( -4, 5 , 83 )   ( 9684, 5 , 83 )  p 2 q 6 r 2 s 12
    122  ( 6, -5 , 61 )   ( 6, 8179 , 61 )  p 4 q 6 r 11 s 11
    125  ( -37, 59 , 105 )   ( 20537, 59 , 105 )  p 10 q 3 r 8 s 12
    145  ( 7, -5 , 19 )   ( 7, 3775 , 19 )  p 0 q 7 r 12 s 12
    149  ( -19, 23 , 449 )   ( 70347, 23 , 449 )  p 2 q 7 r 2 s 14
    170  ( -15, 19 , 82 )   ( 17185, 19 , 82 )  p 5 q 7 r 1 s 15
    173  ( -23, 31 , 97 )   ( 22167, 31 , 97 )  p 10 q 5 r 10 s 14
    190  ( 5, -4 , 23 )   ( 5, 5324 , 23 )  p 0 q 8 r 9 s 15
    194  ( -11, 13 , 292 )   ( 59181, 13 , 292 )  p 2 q 8 r 2 s 16
    197  ( -61, 159 , 101 )   ( 51281, 159 , 101 )  p 14 q 1 r 4 s 16
    209  ( -97, 119 , 611 )   ( 152667, 119 , 611 )  p 8 q 7 r 8 s 16
    226  ( 8, -7 , 113 )   ( 8, 27353 , 113 )  p 6 q 8 r 15 s 15
    242  ( 31, -24 , 115 )   ( 31, 35356 , 115 )  p 1 q 9 r 14 s 16
    245  ( -25, 29 , 737 )   ( 187695, 29 , 737 )  p 2 q 9 r 2 s 18
    257  ( 131, -109 , 755 )   ( 131, 227811 , 755 )  p 4 q 9 r 16 s 16
    269  ( -79, 123 , 227 )   ( 94229, 123 , 227 )  p 14 q 5 r 10 s 18
    290  ( 9, -8 , 145 )   ( 9, 44668 , 145 )  p 7 q 9 r 17 s 17
    302  ( -7, 8 , 227 )   ( 70977, 8 , 227 )  p 2 q 10 r 2 s 20
    305  ( -55, 69 , 293 )   ( 110465, 69 , 293 )  p 8 q 9 r 4 s 20
    314  ( 43, -38 , 469 )   ( 43, 160806 , 469 )  p 4 q 10 r 13 s 19
    325  ( -107, 199 , 235 )   ( 141157, 199 , 235 )  p 18 q 1 r 18 s 18
    334  ( -11, 13 , 82 )   ( 31741, 13 , 82 )  p 6 q 10 r 3 s 21
    362  ( 27, -23 , 178 )   ( 27, 74233 , 178 )  p 1 q 11 r 11 s 21
    365  ( -31, 35 , 1097 )   ( 413211, 35 , 1097 )  p 2 q 11 r 2 s 22
    398  ( -14, 19 , 55 )   ( 29466, 19 , 55 )  p 10 q 10 r 1 s 23
    401  ( -79, 101 , 381 )   ( 193361, 101 , 381 )  p 16 q 7 r 20 s 20
    410  ( -59, 67 , 610 )   ( 277629, 67 , 610 )  p 7 q 11 r 7 s 23
    434  ( -17, 19 , 652 )   ( 291231, 19 , 652 )  p 2 q 12 r 2 s 24
    437  ( -121, 179 , 381 )   ( 244841, 179 , 381 )  p 14 q 9 r 4 s 24
    442  ( -34, 41 , 215 )   ( 113186, 41 , 215 )  p 9 q 11 r 6 s 24
    469  ( -137, 211 , 397 )   ( 285289, 211 , 397 )  p 18 q 7 r 12 s 24
    482  ( -4, 5 , 21 )   ( 12536, 5 , 21 )  p 11 q 11 r 7 s 25
    485  ( -481, 905 , 1037 )   ( 942351, 905 , 1037 )  p 22 q 1 r 22 s 22
    497  ( -313, 407 , 1403 )   ( 899883, 407 , 1403 )  p 16 q 9 r 16 s 24
    509  ( -37, 41 , 1529 )   ( 799167, 41 , 1529 )  p 2 q 13 r 2 s 26
    514  ( 44, -37 , 251 )   ( 44, 151667 , 251 )  p 3 q 13 r 18 s 24
    530  ( 151, -125 , 772 )   ( 151, 489315 , 772 )  p 5 q 13 r 23 s 23
    554  ( -29, 33 , 274 )   ( 170107, 33 , 274 )  p 7 q 13 r 5 s 27
    557  ( -283, 347 , 1613 )   ( 1092003, 347 , 1613 )  p 14 q 11 r 14 s 26
    577  ( -191, 361 , 409 )   ( 444481, 361 , 409 )  p 24 q 1 r 24 s 24
    590  ( -10, 11 , 443 )   ( 267870, 11 , 443 )  p 2 q 14 r 2 s 28
    602  ( 61, -50 , 291 )   ( 61, 211954 , 291 )  p 4 q 14 r 23 s 25
    605  ( -81, 95 , 593 )   ( 416321, 95 , 593 )  p 10 q 13 r 8 s 28
    626  ( 13, -12 , 313 )   ( 13, 204088 , 313 )  p 11 q 13 r 25 s 25
    629  ( -511, 743 , 1661 )   ( 1512627, 743 , 1661 )  p 22 q 7 r 22 s 26
    674  ( 133, -116 , 997 )   ( 133, 761736 , 997 )  p 1 q 15 r 13 s 29
    677  ( -43, 47 , 2033 )   ( 1408203, 47 , 2033 )  p 2 q 15 r 2 s 30
    685  ( -191, 283 , 595 )   ( 601621, 283 , 595 )  p 18 q 11 r 6 s 30
    689  ( 101, -87 , 677 )   ( 101, 536129 , 677 )  p 4 q 15 r 20 s 28
    701  ( -129, 161 , 671 )   ( 583361, 161 , 671 )  p 14 q 13 r 10 s 30
    722  ( -140, 163 , 1063 )   ( 885312, 163 , 1063 )  p 7 q 15 r 1 s 31
    725  ( -211, 323 , 615 )   ( 680261, 323 , 615 )  p 22 q 9 r 14 s 30
    730  ( 14, -13 , 365 )   ( 14, 276683 , 365 )  p 12 q 14 r 27 s 27
    770  ( -23, 25 , 1156 )   ( 909393, 25 , 1156 )  p 2 q 16 r 2 s 32
    773  ( -71, 85 , 451 )   ( 414399, 85 , 451 )  p 10 q 15 r 4 s 32
    785  ( -235, 653 , 369 )   ( 802505, 653 , 369 )  p 28 q 1 r 8 s 32
    794  ( -47, 54 , 391 )   ( 353377, 54 , 391 )  p 11 q 15 r 10 s 32
    830  ( -9, 10 , 103 )   ( 93799, 10 , 103 )  p 8 q 16 r 7 s 33
    842  ( 15, -14 , 421 )   ( 15, 367126 , 421 )  p 13 q 15 r 29 s 29
    845  ( -15, 19 , 73 )   ( 77755, 19 , 73 )  p 22 q 11 r 26 s 30
    869  ( -49, 53 , 2609 )   ( 2313327, 53 , 2609 )  p 2 q 17 r 2 s 34
    874  ( 41, -37 , 434 )   ( 41, 415187 , 434 )  p 3 q 17 r 15 s 33
    890  ( 97, -89 , 1330 )   ( 97, 1270119 , 1330 )  p 5 q 17 r 17 s 33
    901  ( 181, -149 , 871 )   ( 181, 948001 , 871 )  p 6 q 17 r 30 s 30
    917  ( -859, 1415 , 2201 )   ( 3316731, 1415 , 2201 )  p 26 q 9 r 14 s 34
    962  ( -65, 76 , 471 )   ( 526279, 76 , 471 )  p 14 q 16 r 13 s 35
    965  ( 245, -223 , 2879 )   ( 245, 3014883 , 2879 )  p 10 q 17 r 28 s 32
    973  ( -61, 155 , 101 )   ( 249149, 155 , 101 )  p 30 q 5 r 0 s 36
    974  ( -13, 14 , 731 )   ( 725643, 14 , 731 )  p 2 q 18 r 2 s 36
    989  ( -277, 411 , 857 )   ( 1254329, 411 , 857 )  p 22 q 13 r 8 s 36


Mon Jul  6 19:11:55 PDT 2020

Question $2.$

$$\frac{a^2+b^2+c^2}{bc+ca+ab}=k\tag{1}$$
We can get a primitive parametric solution from a known solution below.

Let ${p,q,r}$ is a known solution for equation $(1)$.
Substitute $a=pt+m, b=qt+n, c=rt+s$ to equation $(1)$, then we get
$$t = \frac{-(-m^2+kmn+ksm+kns-s^2-n^2)}{-2nq-2mp+kmq+kpn+knr+kqs+ksp+krm-2sr}$$
Then we get a parametric solution.

$a = (-p+kr+kq)m^2+((-2q+kr)n+(-2r+kq)s)m+pn^2-pkns+ps^2$
$b = m^2q+((-2p+kr)n-kqs)m+(kr-q+kp)n^2+(-2r+kp)sn+qs^2$
$c = rm^2+(-knr+(-2p+kq)s)m+n^2r+(kp-2q)sn+(kp-r+kq)s^2$

$m,n,s$ are arbitrary.

Example:
$(k,p,q,r)=(5,3,5,41)$

$a = 227m^2-15ns+3s^2+3n^2+195mn-57sm$
$b = 5m^2-25sm+5s^2+215n^2+199mn-67ns$
$c = 41m^2-205mn-s^2+41n^2+5ns+19sm$

[$a,b,c$]

[$ 3, 5, 41$]
[$ 3, 5045, 1049$]
[$ 227, 5, 41$]
[$ 17, 5, 111$]
[$ 635, 3149, 17$]
[$ 545, 2901, 47$]
[$ 461, 2663, 75$]
[$ 383, 2435, 101$]
[$1277, 6375, 41$]
[$ 797, 5015, 201$]
[$ 593, 4395, 269$]
[$1361, 8517, 335$]
[$1223, 8105, 381$]
[$1091, 7703, 425$]
[$ 965, 7311, 467$]
[$ 731, 6557, 545$]
[$1739, 11933, 615$]
[$1427, 10965, 719$]
[$1139, 10037, 815$]
[$ 635, 111, 17$]
[$ 545, 59, 47$]
[$1623, 185, 131$]
[$3713, 635, 111$]
[$3491, 503, 185$]
[$3275, 381, 257$]
[$3065, 269, 327$]
[$2861, 167, 395$]
[$5393, 5, 1119$]
[$6653, 1335, 41$]
[$6065, 971, 237$]
[$5501, 647, 425$]
[$8643, 1175, 521$]
[$8301, 983, 635$]
[$7635, 629, 857$]
[$7311, 467, 965$]
[$10727, 75, 2141$]
[$12491, 1853, 615$]
[$11675, 1389, 887$]
[$10883, 965, 1151$]
[$11399, 2217, 125$]
[$11009, 1973, 255$]