Theorem: Pandigital Sum whose Components are Multiples

The following sums use each of the digits $1$ to $9$ exactly once each, while one summand equals $2$ times the other summand:
{{begin-eqn}}
{{eqn | l = 192 + 384
      | r = 576
      | c = where $2 \times 192 = 384$ and so $3 \times 192 = 576$
}}
{{eqn | l = 219 + 438
      | r = 657
      | c = where $2 \times 219 = 438$ and so $3 \times 219 = 657$
}}
{{eqn | l = 273 + 546
      | r = 819
      | c = where $2 \times 273 = 546$ and so $3 \times 273 = 819$
}}
{{eqn | l = 327 + 654
      | r = 981
      | c = where $2 \times 327 = 654$ and so $3 \times 327 = 981$
}}
{{end-eqn}}
{{OEIS|A039667}}
It is interesting to note how the $4$ sums can be divided into two pairs in which each element of the sum is an anagram of its corresponding element in the other.

Proof:
We establish some parameters for the number $x$ such that the equation $x + 2 x = 3 x$ is pandigital.
Firstly, among the three numbers, there cannot be repeating digits or $0$.
If $5 \divides x$:
:$10 \divides 2 x$ and $2 x$ ends in $0$.
By Divisibility by 5, $x$ cannot end in $5$.
Secondly, $x$ must be divisible by $3$.
To see this, consider the concatenation of the three numbers, $\sqbrk {x || 2 x || 3 x}$.
Since the digits of this number is a permutation of $\set {1, \dots, 9}$, they sum to $45$.
Since $9 \divides 45$, by Divisibility by 9:
:$9 \divides \sqbrk {x || 2 x || 3 x}$
We have:
{{begin-eqn}}
{{eqn | l = \sqbrk {x \vert \vert 2 x \vert \vert 3 x}
      | r = 10^6 x + 10^3 \paren {2 x} + 3 x
}}
{{eqn | o = \equiv
      | r = x + 2 x + 3 x
      | rr = \pmod 9
      | c = Congruence of Powers
}}
{{eqn | o = \equiv
      | r = 6 x
      | rr = \pmod 9
}}
{{end-eqn}}
Hence $9 \divides 6 x$.
This gives $3 \divides 2 x$.
By Euclid's Lemma:
:$3 \divides x$
Finally, as $3 x < 1000$, we must have $100 \le x \le 333$.
Putting repeating digits and zeroes into consideration, we list and check all possible candidates from $123$ to $327$:
{{begin-eqn}}
{{eqn | ll = 12X:
      | l = 12X \times 2
      | r = 2XX
      | c = $2$ is repeated
}}
{{eqn | ll = 132:
      | l = 132 \times 2
      | r = 264
      | c = $2$ is repeated
}}
{{eqn | ll = 138:
      | l = 138 \times 3
      | r = 414
      | c = $4$ is repeated
}}
{{eqn | ll = 147:
      | l = 147 \times 2
      | r = 294
      | c = $4$ is repeated
}}
{{eqn | ll = 153:
      | l = 153 \times 2
      | r = 306
      | c = $3$ is repeated
}}
{{eqn | ll = 156:
      | l = 156 \times 2
      | r = 312
      | c = $1$ is repeated
}}
{{eqn | ll = 159:
      | l = 159 \times 2
      | r = 318
      | c = $1$ is repeated
}}
{{eqn | ll = 162:
      | l = 162 \times 2
      | r = 324
      | c = $2$ is repeated
}}
{{eqn | ll = 168:
      | l = 168 \times 2
      | r = 336
      | c = $3, 6$ are repeated
}}
{{eqn | ll = 174:
      | l = 174 \times 2
      | r = 348
      | c = $4$ is repeated
}}
{{eqn | ll = 183:
      | l = 183 \times 2
      | r = 366
      | c = $3, 6$ are repeated
}}
{{eqn | ll = 186:
      | l = 186 \times 3
      | r = 558
      | c = $5, 8$ are repeated
}}
{{eqn | ll = 189:
      | l = 189 \times 2
      | r = 378
      | c = $8$ is repeated
}}
{{eqn | ll = 192:
      | l = 192 + 384
      | r = 576
      | c = is the $1$st solution
}}
{{eqn | ll = 198:
      | l = 198 \times 2
      | r = 396
      | c = $9$ is repeated
}}
{{eqn | ll = 213:
      | l = 213 \times 2
      | r = 426
      | c = $2$ is repeated
}}
{{eqn | ll = 216:
      | l = 216 \times 2
      | r = 432
      | c = $2$ is repeated
}}
{{eqn | ll = 219:
      | l = 219 + 438
      | r = 657
      | c = is the $2$nd solution
}}
{{eqn | ll = 231:
      | l = 231 \times 2
      | r = 462
      | c = $2$ is repeated
}}
{{eqn | ll = 234:
      | l = 234 \times 2
      | r = 468
      | c = $4$ is repeated
}}
{{eqn | ll = 237:
      | l = 237 \times 2
      | r = 474
      | c = $4, 7$ are repeated
}}
{{eqn | ll = 24X:
      | l = 24X \times 2
      | r = 4XX
      | c = $4$ is repeated
}}
{{eqn | ll = 258:
      | l = 258 \times 2
      | r = 516
      | c = $5$ is repeated
}}
{{eqn | ll = 261:
      | l = 261 \times 2
      | r = 522
      | c = $2$ is repeated
}}
{{eqn | ll = 264:
      | l = 264 \times 2
      | r = 528
      | c = $2$ is repeated
}}
{{eqn | ll = 267:
      | l = 267 \times 3
      | r = 801
      | c = $0$ is used
}}
{{eqn | ll = 273:
      | l = 273 + 546
      | r = 819
      | c = is the $3$rd solution
}}
{{eqn | ll = 276:
      | l = 276 \times 2
      | r = 552
      | c = $2, 5$ are repeated
}}
{{eqn | ll = 279:
      | l = 279 \times 3
      | r = 837
      | c = $7$ is repeated
}}
{{eqn | ll = 291:
      | l = 291 \times 2
      | r = 582
      | c = $2$ is repeated
}}
{{eqn | ll = 294:
      | l = 294 \times 2
      | r = 588
      | c = $8$ is repeated
}}
{{eqn | ll = 297:
      | l = 297 \times 2
      | r = 594
      | c = $9$ is repeated
}}
{{eqn | ll = 312:
      | l = 312 \times 2
      | r = 624
      | c = $2$ is repeated
}}
{{eqn | ll = 318:
      | l = 318 \times 2
      | r = 636
      | c = $3, 6$ are repeated
}}
{{eqn | ll = 321:
      | l = 321 \times 2
      | r = 642
      | c = $2$ is repeated
}}
{{eqn | ll = 324:
      | l = 324 \times 2
      | r = 648
      | c = $4$ is repeated
}}
{{eqn | ll = 327:
      | l = 327 + 654
      | r = 981
      | c = is the $4$th solution
}}
{{end-eqn}}
Hence the result.
{{qed}}