CJ7

Let's call numbers that are multiples of 7 and don't contain the digit $7$ unseveners .

First notice that for any non-negative arithmetic sequences with common difference of 7, their units digit cycles through $7, 4, 1, 8, 5, 2, 9, 6, 3, 0$ .

We see that the numbers 0 to 9 all appear once in the units digit of any 10 consecutive terms.

Lemma 1: For any non-negative arithmetic sequences with common difference of 7, the units digit of every 10 consecutive terms must have every number from $0$ to $9$ .

Hence, if we consider all non-negative multiples of 7 smaller than 70: $0,\, \color{#EC7300}7\color{#333333},\, 14,\, 21,\, 28,\, 35,\, 42,\, 49,\, 56,\, 63$ by lemma 1 , and because no numbers have a $7$ in the tens place, we know that exactly $9$ out of these $10$ are unseveners .

This can be generalized to any 10 consecutive multiples of 7, as long as none of them have a $7$ in any other place other than the units.

Lemma 2: For any 10 consecutive multiples of 7, where none of them have a $7$ in any place except the units, exactly $9$ of them are unseveners .

Now consider all non-negative multiples of 7 smaller than 700, we write them in a $10\times10$ array like this: $\begin{matrix} 0 & \color{#EC7300}70 & 140 & 210 & 280 & 350 & 420 & 490 & 560 & 630\\ \color{#EC7300}7 & \color{#EC7300}77 & \color{#EC7300}147 & \color{#EC7300}217 & \color{#EC7300}287 & \color{#EC7300}357 & \color{#EC7300}427 & \color{#EC7300}497 & \color{#EC7300}567 & \color{#EC7300}637\\ 14 & 84 & 154 & 224 & 294 & 364 & 434 & 504 & \color{#EC7300}574 & 644\\ 21 & 91 & 161 & 231 & 301 & \color{#EC7300}371 & 441 & 511 & 581 & 651\\ 28 & 98 & 168 & 238 & 308 & \color{#EC7300}378 & 448 & 518 & 588 & 658\\ 35 & 105 & \color{#EC7300}175 & 245 & 315 & 385 & 455 & 525 & 595 & 665\\ 42 & 112 & 182 & 252 & 322 & 392 & 462 & 532 & 602 & \color{#EC7300}672\\ 49 & 119 & 189 & 259 & 329 & 399 & 469 & 539 & 609 & \color{#EC7300}679\\ 56 & 126 & 196 & 266 & 336 & 406 & \color{#EC7300}476 & 546 & 616 & 686\\ 63 & 133 & 203 & \color{#EC7300}273 & 343 & 413 & 483 & 553 & 623 & 693\\ \end{matrix}$

1 out of 10 multiples of 7 have $7$ in the units place, hence we must have 10 out of 100 multiples of 7 that have $7$ in the units place, this is represented by the second row.

As for the remaining 9 rows, if we ignore their units digit, the remaining digits form a non-negative arithmetic sequence with a common difference of 7 consisting of 10 terms, by lemma 1 , and because there are no numbers with a $7$ in its hundreds place, we know that there must be exactly 9 unseveners in each row.

Thus, for multiples of 7 smaller than 700, exactly $9\times9=81$ of them are unseveners .

This can be generalized to any 100 consecutive multiples of 7, as long as none of them have a $7$ in any other place other than the units and tens.

Lemma 3: For any 100 consecutive multiples of 7, where none of them have a $7$ in any place except the units and the tens, exactly $81$ of them are unseveners .

Now consider all non-negative multiples of 7 smaller than 7000, imagine writing them in a 3D $10\times10\times10$ array like so (each subsequent 'plane' of numbers is 700 more than their respective numbers in the previous plane):

$100-81=19$ out of 100 multiples of 7 have at least one $7$ in their units and tens place, hence we have 190 out of 1000 multiples of 7 that have at least one $7$ in their units and tens place, this is represented by the 19 "z-rows" .

As for the remaining $81$ z-rows , if we ignore both their units and tens digit, the remaining digits form an arithmetic sequence of common difference 7 with 10 terms, by lemma 1 , and because there are no numbers with a $7$ in its thousands place, we know that there must be exactly 9 unseveners in each z-row .

Thus, for multiples of 7 smaller than 7000, $81\times9=729$ of them are unseveners .

We now know that from 1 to 6999, $729$ of them are unseveners . But our target range is 1000 to 9999, we'll do the remaining inclusion and exclusion manually.

1 to 999:

We already know that from 1 to 699, $81$ of them are unseveners . From 700 to 799, there are no unseveners . By lemma 2 , there are $9$ unseveners from 800 to 869, and $9$ in 880 to 949. Plus the remaining unseveners $952$ , $959$ , $966$ , $980$ , $994$ , making a total of $104$ unseveners .

7000 to 7999:

There are no unseveners in this range.

8000 to 8999:

By lemma 3 , there are $81$ seveners from 8000 to 8699. From 8700 to 8799, there are no unseveners . By lemma 2 , there are $9$ unseveners from 8800 to 8869, and $9$ from 8880 to 8949. Plus the remaining unseveners $8953$ , $8960$ , $8981$ , $8988$ , $8995$ , making a total of $104$ unseveners .

9000 to 9999:

By lemma 3 , there are $81$ seveners from 9000 to 9699. From 9700 to 9799, there are no unseveners . By lemma 2 , there are $9$ unseveners from 9800 to 9869, and $9$ in 9880 to 9949. Plus the remaining unseveners $9954$ , $9961$ , $9968$ , $9982$ , $9989$ , $9996$ , making a total of $105$ seveners .

Therefore, the total number of unseveners from 1000 to 9999 is $729-104+104+105=834$ .

$\text{Length}[\text{Table}[\text{If}[\neg \text{MemberQ}[\text{IntegerDigits}[i],7],i,\text{Nothing}],\{i,1001,9999,7\}]] \Longrightarrow 834$