## 1. Covering set 被覆集合

Some near-repdigit-related sequences do not include prime numbers because those elements are always divisible by at least one element of the corresponding covering set. 一部のニアレプディジット関連の数列は、そのすべての要素が対応する被覆集合の少なくとも 1 つの要素で割り切れるため、素数を含みません。

## 2. Examples 例

### 2.1. 91w3

Quasi-repdigit sequence $91{\w}3 = \{ 93, 913, 9113, 91113, 911113, \cdots \} = (82 \times 10^n+17)/9$ is entirely-covered by the covering set $\{ 3, 7, 11, 13 \}$. クワージレプディジットの数列 $91{\w}3 = \{ 93, 913, 9113, 91113, 911113, \cdots \} = (82 \times 10^n+17)/9$ は被覆集合 $\{ 3, 7, 11, 13 \}$ で完全に覆われます。

\begin{align} (82 \times 10^{3k+1}+17)/9 & = 3 \times \Bigl[31+30340 \times (10^{3k}-1)/(10^3-1)\Bigr] \\ & = 3 \times \biggl[31+30340 \times \sum_{m=0}^{k-1}10^{3m}\biggr] \\ (82 \times 10^{6k+5}+17)/9 & = 7 \times \Bigl[130159+130158600000 \times (10^{6k}-1)/(10^6-1)\Bigr] \\ & = 7 \times \biggl[130159+130158600000 \times \sum_{m=0}^{k-1}10^{6m}\biggr] \\ (82 \times 10^{2k}+17)/9 & = 11 \times \Bigl[1+82 \times (10^{2k}-1)/(10^2-1)\Bigr] \\ & = 11 \times \biggl[1+82 \times \sum_{m=0}^{k-1}10^{2m}\biggr] \\ (82 \times 10^{6k+3}+17)/9 & = 13 \times \Bigl[701+700854000 \times (10^{6k}-1)/(10^6-1)\Bigr] \\ & = 13 \times \biggl[701+700854000 \times \sum_{m=0}^{k-1}10^{6m}\biggr] \end{align} $$\begin{array}{|c|cccccc|} \hline n & 6k & 6k+1 & 6k+2 & 6k+3 & 6k+4 & 6k+5 \\ \hline 3 & \cdot & 3k+1 & \cdot & \cdot & 3k+1 & \cdot \\ 7 & \cdot & \cdot & \cdot & \cdot & \cdot & 6k+5 \\ 11 & 2k & \cdot & 2k & \cdot & 2k & \cdot \\ 13 & \cdot & \cdot & \cdot & 6k+3 & \cdot & \cdot \\ \hline (82 \times 10^n+17)/9 \text{ is divisible by} & 11 & 3 & 11 & 13 & 3,11 & 7 \\ \hline \end{array}$$

### 2.2. 381w

Quasi-repdigit sequence $381{\w} = \{ 38, 381, 3811, 38111, 381111, \cdots \} = (343 \times 10^n-1)/9$ is entirely-covered by the covering set $\{ 3, 37, 23{\w} \}$. クワージレプディジットの数列 $381{\w} = \{ 38, 381, 3811, 38111, 381111, \cdots \} = (343 \times 10^n-1)/9$ は被覆集合 $\{ 3, 37, 23{\w} \}$ で完全に覆われます。

Note: $23{\w} = \{ 2, 23, 233, 2333, \cdots \} = (7 \times 10^k-1)/3$

\begin{align} (343 \times 10^{3k+1}-1)/9 & = 3 \times \Bigl[127+126910 \times (10^{3k}-1)/(10^3-1)\Bigr] \\ & = 3 \times \biggl[127+126910 \times \sum_{m=0}^{k-1}10^{3m}\biggr] \\ (343 \times 10^{3k+2}-1)/9 & = 37 \times \Bigl[103+102900 \times (10^{3k}-1)/(10^3-1)\Bigr] \\ & = 37 \times \biggl[103+102900 \times \sum_{m=0}^{k-1}10^{3m}\biggr] \\ (343 \times 10^{3k}-1)/9 & = (7 \times 10^k-1)/3 \times (49 \times 10^{2k}+7 \times 10^k+1)/3 \\ & = 23{\w} \times 163{\w}56{\w}7 \end{align} $$\begin{array}{|c|ccc|} \hline n & 3k & 3k+1 & 3k+2 \\ \hline 3 & \cdot & 3k+1 & \cdot \\ 37 & \cdot & \cdot & 3k+2 \\ 23{\w} & 3k & \cdot & \cdot \\ \hline (343 \times 10^n-1)/9 \text{ is divisible by} & 23{\w} & 3 & 37 \\ \hline \end{array}$$

### 2.3. 13266340w1

Near-repdigit-related sequence $13266340{\w}1 = \{ 13266341, 132663401, 1326634001, 13266340001, 132663400001, \cdots \} = 1326634 \times 10^n+1$ is entirely-covered by the covering set $\{ 11, 13, 37, 101, 9901 \}$. ニアレプディジット関連の数列 $13266340{\w}1 = \{ 13266341, 132663401, 1326634001, 13266340001, 132663400001, \cdots \} = 1326634 \times 10^n+1$ は被覆集合 $\{ 11, 13, 37, 101, 9901 \}$ で完全に覆われます。

\begin{align} 1326634 \times 10^{2k+1}+1 & = 11 \times \Bigl[1206031+119397060 \times (10^{2k}-1)/(10^2-1)\Bigr] \\ & = 11 \times \biggl[1206031+119397060 \times \sum_{m=0}^{k-1}10^{2m}\biggr] \\ 1326634 \times 10^{6k+2}+1 & = 13 \times \Bigl[10204877+10204866718200 \times (10^{6k}-1)/(10^6-1)\Bigr] \\ & = 13 \times \biggl[10204877+10204866718200 \times \sum_{m=0}^{k-1}10^{6m}\biggr] \\ 1326634 \times 10^{3k}+1 & = 37 \times \Bigl[35855+35819118 \times (10^{3k}-1)/(10^3-1)\Bigr] \\ & = 37 \times \biggl[35855+35819118 \times \sum_{m=0}^{k-1}10^{3m}\biggr] \\ 1326634 \times 10^{4k}+1 & = 101 \times \Bigl[13135+131336766 \times (10^{4k}-1)/(10^4-1)\Bigr] \\ & = 101 \times \biggl[13135+131336766 \times \sum_{m=0}^{k-1}10^{4m}\biggr] \\ 1326634 \times 10^{12k+10}+1 & = 9901 \times \Bigl[1339899000101+1339899000099660000000000 \times (10^{12k}-1)/(10^{12}-1)\Bigr] \\ & = 9901 \times \biggl[1339899000101+1339899000099660000000000 \times \sum_{m=0}^{k-1}10^{12m}\biggr] \end{align} $$\begin{array}{|c|cccccc|} \hline n & 12k & 12k+1 & 12k+2 & 12k+3 & 12k+4 & 12k+5 & 12k+6 & 12k+7 & 12k+8 & 12k+9 & 12k+10 & 12k+11 \\ \hline 11 & \cdot & 2k+1 & \cdot & 2k+1 & \cdot & 2k+1 & \cdot & 2k+1 & \cdot & 2k+1 & \cdot & 2k+1 \\ 13 & \cdot & \cdot & 6k+2 & \cdot & \cdot & \cdot & \cdot & \cdot & 6k+2 & \cdot & \cdot & \cdot \\ 37 & 3k & \cdot & \cdot & 3k & \cdot & \cdot & 3k & \cdot & \cdot & 3k & \cdot & \cdot \\ 101 & 4k & \cdot & \cdot & \cdot & 4k & \cdot & \cdot & \cdot & 4k & \cdot & \cdot & \cdot \\ 9901 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 12k+10 & \cdot \\ \hline 1326634 \times 10^n+1 \text{ is divisible by} & 37,101 & 11 & 13 & 11,37 & 101 & 11 & 37 & 11 & 13,101 & 11,37 & 9901 & 11 \\ \hline \end{array}$$

## 3. Covering set of near-repdigit-related sequences ニアレプディジット関連の数列の被覆集合

There are countless entirely-covered near-repdigit-related sequences. The following table shows one near-repdigit-related sequence for each factor sequence. 完全に覆われるニアレプディジット関連の数列は無数にあります。以下の表は因数列毎に 1 つのニアレプディジット関連の数列を示しています。

Covering set of near-repdigit-related sequences ニアレプディジット関連の数列の被覆集合
flabelwlabelgeneral term 一般項covering set 被覆集合cycle 周期factor sequence 因数列
38111381w(343·10n-1)/9{3,37,23w}323w,3,37
110731w073(10n-343)/9{3,37,3w1}33w1,37,3
3220132w01(29·10n-191)/9{3,7,11,13}63,11,7,11,13,11
9444994w9(85·10n+41)/9{3,7,11,13}67,11,13,11,3,11
4115141w51(37·10n+359)/9{3,7,11,13}611,3,11,7,11,13
9111391w3(82·10n+17)/9{3,7,11,13}611,3,11,13,11,7
6552365w23(59·10n-293)/9{3,7,11,13}613,11,7,11,3,11
8115181w51(73·10n+359)/9{3,7,11,37}63,11,37,11,7,11
4220742w07(38·10n-137)/9{3,7,11,37}611,3,11,7,11,37
444074w07(4·10n-337)/9{3,7,11,37}637,11,7,11,3,11
555375w37(5·10n-167)/9{3,7,13,37}63,13,37,3,7,37
9771797w17(88·10n-547)/9{3,7,13,37}63,37,7,3,37,13
4880348w03(44·10n-773)/9{3,7,13,37}63,37,13,3,37,7
37111371w(334·10n-1)/9{3,7,13,37}637,7,3,37,13,3
-271469w27147·10n-1{7,11,13,37}67,11,13,11,37,11
-93510w19351·10n+1{7,11,13,37}67,11,37,11,13,11
-360479w36048·10n-1{7,11,13,37}611,7,11,13,11,37
-101759w10176·10n-1{7,11,13,37}611,7,11,37,11,13
-176760w117676·10n+1{7,11,13,37}611,13,11,7,11,37
-248260w124826·10n+1{7,11,13,37}611,13,11,37,11,7
-359159w35916·10n-1{7,11,13,37}611,37,11,7,11,13
-176009w17601·10n-1{7,11,13,37}611,37,11,13,11,7
-286120w128612·10n+1{7,11,13,37}613,11,7,11,37,11
-260380w126038·10n+1{7,11,13,37}613,11,37,11,7,11
-258269w25827·10n-1{7,11,13,37}637,11,7,11,13,11
-91750w19175·10n+1{7,11,13,37}637,11,13,11,7,11
-44596649w4459665·10n-1{11,73,101,137}811,73,11,101,11,137,11,101
-94890409w9489041·10n-1{11,73,101,137}811,101,11,73,11,101,11,137
-95890510w19589051·10n+1{11,73,101,137}811,101,11,137,11,101,11,73
-33495540w13349554·10n+1{11,73,101,137}811,137,11,101,11,73,11,101
-109589050w110958905·10n+1{11,73,101,137}873,11,101,11,137,11,101,11
-51095890w15109589·10n+1{11,73,101,137}8101,11,73,11,101,11,137,11
-41094889w4109489·10n-1{11,73,101,137}8101,11,137,11,101,11,73,11
-1622070w1162207·10n+1{11,73,101,137}8137,11,101,11,73,11,101,11
-1313368639w131336864·10n-1{7,11,13,101,9901}127,11,101,11,13,11,101,11,9901,11,101,11
-5936739610w1593673961·10n+1{7,11,13,101,9901}127,11,101,11,9901,11,101,11,13,11,101,11
-6137342869w613734287·10n-1{7,11,13,101,9901}1211,7,11,101,11,13,11,101,11,9901,11,101
-407327039w40732704·10n-1{7,11,13,101,9901}1211,7,11,101,11,9901,11,101,11,13,11,101
-1297338029w129733803·10n-1{7,11,13,101,9901}1211,13,11,101,11,7,11,101,11,9901,11,101
-2057328689w205732869·10n-1{7,11,13,101,9901}1211,13,11,101,11,9901,11,101,11,7,11,101
-1362674729w136267473·10n-1{7,11,13,101,9901}1211,101,11,7,11,101,11,13,11,101,11,9901
-396327130w139632713·10n+1{7,11,13,101,9901}1211,101,11,7,11,101,11,9901,11,101,11,13
-287326920w128732692·10n+1{7,11,13,101,9901}1211,101,11,13,11,101,11,7,11,101,11,9901
-6886333620w1688633362·10n+1{7,11,13,101,9901}1211,101,11,13,11,101,11,9901,11,101,11,7
-4477341210w1447734121·10n+1{7,11,13,101,9901}1211,101,11,9901,11,101,11,7,11,101,11,13
-692663959w69266396·10n-1{7,11,13,101,9901}1211,101,11,9901,11,101,11,13,11,101,11,7
-2716329449w271632945·10n-1{7,11,13,101,9901}1211,9901,11,101,11,7,11,101,11,13,11,101
-10813684080w11081368408·10n+1{7,11,13,101,9901}1211,9901,11,101,11,13,11,101,11,7,11,101
-2963370289w296337029·10n-1{7,11,13,101,9901}1213,11,101,11,7,11,101,11,9901,11,101,11
-9456743130w1945674313·10n+1{7,11,13,101,9901}1213,11,101,11,9901,11,101,11,7,11,101,11
-3616737289w361673729·10n-1{7,11,13,101,9901}12101,11,7,11,101,11,13,11,101,11,9901,11
-6046738709w604673871·10n-1{7,11,13,101,9901}12101,11,7,11,101,11,9901,11,101,11,13,11
-12883279210w11288327921·10n+1{7,11,13,101,9901}12101,11,13,11,101,11,7,11,101,11,9901,11
-8803276140w1880327614·10n+1{7,11,13,101,9901}12101,11,13,11,101,11,9901,11,101,11,7,11
-5276637949w527663795·10n-1{7,11,13,101,9901}12101,11,9901,11,101,11,7,11,101,11,13,11
-13093380420w11309338042·10n+1{7,11,13,101,9901}12101,11,9901,11,101,11,13,11,101,11,7,11
-2866735530w1286673553·10n+1{7,11,13,101,9901}129901,11,101,11,7,11,101,11,13,11,101,11
-1973269309w197326931·10n-1{7,11,13,101,9901}129901,11,101,11,13,11,101,11,7,11,101,11
-993267319w99326732·10n-1{7,11,37,101,9901}127,11,37,11,9901,11,101,11,37,11,101,11
-9466642140w1946664214·10n+1{7,11,37,101,9901}127,11,101,11,37,11,101,11,9901,11,37,11
-22692696060w12269269606·10n+1{7,11,37,101,9901}1211,7,11,37,11,9901,11,101,11,37,11,101
-12342675610w11234267561·10n+1{7,11,37,101,9901}1211,7,11,101,11,37,11,101,11,9901,11,37
-3257329890w1325732989·10n+1{7,11,37,101,9901}1211,37,11,7,11,101,11,37,11,101,11,9901
-23627360359w2362736036·10n-1{7,11,37,101,9901}1211,37,11,101,11,7,11,37,11,9901,11,101
-7952681320w1795268132·10n+1{7,11,37,101,9901}1211,37,11,101,11,9901,11,37,11,7,11,101
-38422701689w3842270169·10n-1{7,11,37,101,9901}1211,37,11,9901,11,101,11,37,11,101,11,7
-1936328670w1193632867·10n+1{7,11,37,101,9901}1211,101,11,7,11,37,11,9901,11,101,11,37
-38982712349w3898271235·10n-1{7,11,37,101,9901}1211,101,11,37,11,101,11,7,11,37,11,9901
-9196335930w1919633593·10n+1{7,11,37,101,9901}1211,101,11,37,11,101,11,9901,11,37,11,7
-2452665719w245266572·10n-1{7,11,37,101,9901}1211,101,11,9901,11,37,11,7,11,101,11,37
-11143684410w11114368441·10n+1{7,11,37,101,9901}1211,9901,11,37,11,7,11,101,11,37,11,101
-3883677150w1388367715·10n+1{7,11,37,101,9901}1211,9901,11,101,11,37,11,101,11,7,11,37
-24406758079w2440675808·10n-1{7,11,37,101,9901}1237,11,7,11,101,11,37,11,101,11,9901,11
-8353375679w835337568·10n-1{7,11,37,101,9901}1237,11,101,11,7,11,37,11,9901,11,101,11
-5943272269w594327227·10n-1{7,11,37,101,9901}1237,11,101,11,9901,11,37,11,7,11,101,11
-13856646529w1385664653·10n-1{7,11,37,101,9901}1237,11,9901,11,101,11,37,11,101,11,7,11
-37616770279w3761677028·10n-1{7,11,37,101,9901}12101,11,7,11,37,11,9901,11,101,11,37,11
-19456753129w1945675313·10n-1{7,11,37,101,9901}12101,11,37,11,101,11,7,11,37,11,9901,11
-6493273830w1649327383·10n+1{7,11,37,101,9901}12101,11,37,11,101,11,9901,11,37,11,7,11
-3963371290w1396337129·10n+1{7,11,37,101,9901}12101,11,9901,11,37,11,7,11,101,11,37,11
-25966758630w12596675863·10n+1{7,11,37,101,9901}129901,11,37,11,7,11,101,11,37,11,101,11
-18143285479w1814328548·10n-1{7,11,37,101,9901}129901,11,101,11,37,11,101,11,7,11,37,11
-22077348709w2207734871·10n-1{11,13,37,101,9901}1211,13,11,37,11,9901,11,101,11,37,11,101
-52777389509w5277738951·10n-1{11,13,37,101,9901}1211,13,11,101,11,37,11,101,11,9901,11,37
-21162694529w2116269453·10n-1{11,13,37,101,9901}1211,37,11,13,11,101,11,37,11,101,11,9901
-3277340009w327734001·10n-1{11,13,37,101,9901}1211,37,11,101,11,13,11,37,11,9901,11,101
-3882677250w1388267725·10n+1{11,13,37,101,9901}1211,37,11,101,11,9901,11,37,11,13,11,101
-14487351220w11448735122·10n+1{11,13,37,101,9901}1211,37,11,9901,11,101,11,37,11,101,11,13
-42636369370w14263636937·10n+1{11,13,37,101,9901}1211,101,11,13,11,37,11,9901,11,101,11,37
-47122720489w4712272049·10n-1{11,13,37,101,9901}1211,101,11,37,11,101,11,13,11,37,11,9901
-39643712909w3964371291·10n-1{11,13,37,101,9901}1211,101,11,37,11,101,11,9901,11,37,11,13
-35012698279w3501269828·10n-1{11,13,37,101,9901}1211,101,11,9901,11,37,11,13,11,101,11,37
-9206335939w920633594·10n-1{11,13,37,101,9901}1211,9901,11,37,11,13,11,101,11,37,11,101
-32746359479w3274635948·10n-1{11,13,37,101,9901}1211,9901,11,101,11,37,11,101,11,13,11,37
-43776777450w14377677745·10n+1{11,13,37,101,9901}1213,11,37,11,9901,11,101,11,37,11,101,11
-51583418909w5158341891·10n-1{11,13,37,101,9901}1213,11,101,11,37,11,101,11,9901,11,37,11
-13266340w11326634·10n+1{11,13,37,101,9901}1237,11,13,11,101,11,37,11,101,11,9901,11
-66993333319w6699333332·10n-1{11,13,37,101,9901}1237,11,101,11,9901,11,37,11,13,11,101,11
-66766699439w6676669944·10n-1{11,13,37,101,9901}1237,11,9901,11,101,11,37,11,101,11,13,11
-49826782489w4982678249·10n-1{11,13,37,101,9901}12101,11,13,11,37,11,9901,11,101,11,37,11
-4963271290w1496327129·10n+1{11,13,37,101,9901}12101,11,37,11,101,11,13,11,37,11,9901,11
-26066758729w2606675873·10n-1{11,13,37,101,9901}12101,11,37,11,101,11,9901,11,37,11,13,11
-32666665339w3266666534·10n-1{11,13,37,101,9901}12101,11,9901,11,37,11,13,11,101,11,37,11
-66666799330w16666679933·10n+1{11,13,37,101,9901}129901,11,37,11,13,11,101,11,37,11,101,11
-10003277339w1000327734·10n-1{11,13,37,101,9901}129901,11,101,11,37,11,101,11,13,11,37,11
-14774658240w11477465824·10n+1{7,11,13,73,137,9901}2411,7,11,137,11,13,11,7,11,9901,11,13,
11,7,11,73,11,13,11,7,11,9901,11,13
-8109503059w810950306·10n-1{7,11,13,73,137,9901}2411,7,11,9901,11,13,11,7,11,73,11,13,
11,7,11,9901,11,13,11,7,11,137,11,13
-59390257510w15939025751·10n+1{7,11,13,73,137,9901}2411,13,11,7,11,9901,11,13,11,7,11,73,
11,13,11,7,11,9901,11,13,11,7,11,137
-66054412590w16605441259·10n+1{7,11,13,73,137,9901}2411,73,11,13,11,7,11,9901,11,13,11,7,
11,137,11,13,11,7,11,9901,11,13,11,7
-49611238720w14961123872·10n+1{7,11,37,73,137,9901}2437,11,9901,11,7,11,37,11,73,11,7,11,
37,11,9901,11,7,11,37,11,137,11,7,11
-39400930480w13940093048·10n+1{7,11,37,73,137,9901}249901,11,7,11,37,11,137,11,7,11,37,11,
9901,11,7,11,37,11,73,11,7,11,37,11
-6208139830w1620813983·10n+1{7,13,37,73,101,137}247,37,13,73,37,101,7,37,13,101,37,73,
7,37,13,137,37,101,7,37,13,101,37,137
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11,101,11,17,11,101,11,137,11,101,11,641,
11,101,11,137,11,101,11,17
Makoto Kamada / March 7, 2015 2015 年 3 月 7 日
-17187514510w11718751451·10n+1{7,11,13,31,41,61,101,211,241}6011,101,11,13,11,101,11,241,11,101,11,41,
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11,101,11,13,11,101,11,41,11,101,11,7,
11,101,11,13,11,101,11,211,11,101,11,7,
11,101,11,41,11,101,11,31,11,101,11,7
Makoto Kamada / February 14, 2015 2015 年 2 月 14 日
-54466351710w15446635171·10n+1{11,13,17,37,73,97,101,641}9611,73,11,37,11,13,11,101,11,37,11,101,
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11,73,11,37,11,13,11,101,11,37,11,101,
11,97,11,37,11,13,11,101,11,37,11,101,
11,73,11,37,11,13,11,101,11,37,11,101,
11,17,11,37,11,13,11,101,11,37,11,101,
11,73,11,37,11,13,11,101,11,37,11,101,
11,641,11,37,11,13,11,101,11,37,11,101
Makoto Kamada / March 22, 2015 2015 年 3 月 22 日
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11,37,11,101,11,13,11,37,11,17,11,101,
11,37,11,101,11,13,11,37,11,137,11,101,
11,37,11,101,11,13,11,37,11,449,11,101,
11,37,11,101,11,13,11,37,11,137,11,101,
11,37,11,101,11,13,11,37,11,17,11,101,
11,37,11,101,11,13,11,37,11,137,11,101
Makoto Kamada / February 19, 2015 2015 年 2 月 19 日