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。
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。
$$\begin{align} (82 \times 10^{3k+1}+17)/9 & = 3 \times \Bigl[31+30340 \times (10^{3k}-1)/(10^3-1)\Bigr] \\ & = 3 \times \biggl[31+30340 \times \sum_{m=0}^{k-1}10^{3m}\biggr] \\ (82 \times 10^{6k+5}+17)/9 & = 7 \times \Bigl[130159+130158600000 \times (10^{6k}-1)/(10^6-1)\Bigr] \\ & = 7 \times \biggl[130159+130158600000 \times \sum_{m=0}^{k-1}10^{6m}\biggr] \\ (82 \times 10^{2k}+17)/9 & = 11 \times \Bigl[1+82 \times (10^{2k}-1)/(10^2-1)\Bigr] \\ & = 11 \times \biggl[1+82 \times \sum_{m=0}^{k-1}10^{2m}\biggr] \\ (82 \times 10^{6k+3}+17)/9 & = 13 \times \Bigl[701+700854000 \times (10^{6k}-1)/(10^6-1)\Bigr] \\ & = 13 \times \biggl[701+700854000 \times \sum_{m=0}^{k-1}10^{6m}\biggr] \end{align}$$ $$\begin{array}{|c|cccccc|} \hline n & 6k & 6k+1 & 6k+2 & 6k+3 & 6k+4 & 6k+5 \\ \hline 3 & \cdot & 3k+1 & \cdot & \cdot & 3k+1 & \cdot \\ 7 & \cdot & \cdot & \cdot & \cdot & \cdot & 6k+5 \\ 11 & 2k & \cdot & 2k & \cdot & 2k & \cdot \\ 13 & \cdot & \cdot & \cdot & 6k+3 & \cdot & \cdot \\ \hline (82 \times 10^n+17)/9 \text{ is divisible by} & 11 & 3 & 11 & 13 & 3,11 & 7 \\ \hline \end{array}$$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: $23{\w} = \{ 2, 23, 233, 2333, \cdots \} = (7 \times 10^k-1)/3$
$$\begin{align} (343 \times 10^{3k+1}-1)/9 & = 3 \times \Bigl[127+126910 \times (10^{3k}-1)/(10^3-1)\Bigr] \\ & = 3 \times \biggl[127+126910 \times \sum_{m=0}^{k-1}10^{3m}\biggr] \\ (343 \times 10^{3k+2}-1)/9 & = 37 \times \Bigl[103+102900 \times (10^{3k}-1)/(10^3-1)\Bigr] \\ & = 37 \times \biggl[103+102900 \times \sum_{m=0}^{k-1}10^{3m}\biggr] \\ (343 \times 10^{3k}-1)/9 & = (7 \times 10^k-1)/3 \times (49 \times 10^{2k}+7 \times 10^k+1)/3 \\ & = 23{\w} \times 163{\w}56{\w}7 \end{align}$$ $$\begin{array}{|c|ccc|} \hline n & 3k & 3k+1 & 3k+2 \\ \hline 3 & \cdot & 3k+1 & \cdot \\ 37 & \cdot & \cdot & 3k+2 \\ 23{\w} & 3k & \cdot & \cdot \\ \hline (343 \times 10^n-1)/9 \text{ is divisible by} & 23{\w} & 3 & 37 \\ \hline \end{array}$$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。
$$\begin{align} 1326634 \times 10^{2k+1}+1 & = 11 \times \Bigl[1206031+119397060 \times (10^{2k}-1)/(10^2-1)\Bigr] \\ & = 11 \times \biggl[1206031+119397060 \times \sum_{m=0}^{k-1}10^{2m}\biggr] \\ 1326634 \times 10^{6k+2}+1 & = 13 \times \Bigl[10204877+10204866718200 \times (10^{6k}-1)/(10^6-1)\Bigr] \\ & = 13 \times \biggl[10204877+10204866718200 \times \sum_{m=0}^{k-1}10^{6m}\biggr] \\ 1326634 \times 10^{3k}+1 & = 37 \times \Bigl[35855+35819118 \times (10^{3k}-1)/(10^3-1)\Bigr] \\ & = 37 \times \biggl[35855+35819118 \times \sum_{m=0}^{k-1}10^{3m}\biggr] \\ 1326634 \times 10^{4k}+1 & = 101 \times \Bigl[13135+131336766 \times (10^{4k}-1)/(10^4-1)\Bigr] \\ & = 101 \times \biggl[13135+131336766 \times \sum_{m=0}^{k-1}10^{4m}\biggr] \\ 1326634 \times 10^{12k+10}+1 & = 9901 \times \Bigl[1339899000101+1339899000099660000000000 \times (10^{12k}-1)/(10^{12}-1)\Bigr] \\ & = 9901 \times \biggl[1339899000101+1339899000099660000000000 \times \sum_{m=0}^{k-1}10^{12m}\biggr] \end{align}$$ $$\begin{array}{|c|cccccc|} \hline n & 12k & 12k+1 & 12k+2 & 12k+3 & 12k+4 & 12k+5 & 12k+6 & 12k+7 & 12k+8 & 12k+9 & 12k+10 & 12k+11 \\ \hline 11 & \cdot & 2k+1 & \cdot & 2k+1 & \cdot & 2k+1 & \cdot & 2k+1 & \cdot & 2k+1 & \cdot & 2k+1 \\ 13 & \cdot & \cdot & 6k+2 & \cdot & \cdot & \cdot & \cdot & \cdot & 6k+2 & \cdot & \cdot & \cdot \\ 37 & 3k & \cdot & \cdot & 3k & \cdot & \cdot & 3k & \cdot & \cdot & 3k & \cdot & \cdot \\ 101 & 4k & \cdot & \cdot & \cdot & 4k & \cdot & \cdot & \cdot & 4k & \cdot & \cdot & \cdot \\ 9901 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 12k+10 & \cdot \\ \hline 1326634 \times 10^n+1 \text{ is divisible by} & 37,101 & 11 & 13 & 11,37 & 101 & 11 & 37 & 11 & 13,101 & 11,37 & 9901 & 11 \\ \hline \end{array}$$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。
| flabel | wlabel | general term 一般項 | covering set 被覆集合 | cycle 周期 | factor sequence 因数列 |
|---|---|---|---|---|---|
| 38111 | 381w | (343·10n-1)/9 | {3,37,23w} | 3 | 23w,3,37 |
| 11073 | 1w073 | (10n-343)/9 | {3,37,3w1} | 3 | 3w1,37,3 |
| 32201 | 32w01 | (29·10n-191)/9 | {3,7,11,13} | 6 | 3,11,7,11,13,11 |
| 94449 | 94w9 | (85·10n+41)/9 | {3,7,11,13} | 6 | 7,11,13,11,3,11 |
| 41151 | 41w51 | (37·10n+359)/9 | {3,7,11,13} | 6 | 11,3,11,7,11,13 |
| 91113 | 91w3 | (82·10n+17)/9 | {3,7,11,13} | 6 | 11,3,11,13,11,7 |
| 65523 | 65w23 | (59·10n-293)/9 | {3,7,11,13} | 6 | 13,11,7,11,3,11 |
| 81151 | 81w51 | (73·10n+359)/9 | {3,7,11,37} | 6 | 3,11,37,11,7,11 |
| 42207 | 42w07 | (38·10n-137)/9 | {3,7,11,37} | 6 | 11,3,11,7,11,37 |
| 44407 | 4w07 | (4·10n-337)/9 | {3,7,11,37} | 6 | 37,11,7,11,3,11 |
| 55537 | 5w37 | (5·10n-167)/9 | {3,7,13,37} | 6 | 3,13,37,3,7,37 |
| 97717 | 97w17 | (88·10n-547)/9 | {3,7,13,37} | 6 | 3,37,7,3,37,13 |
| 48803 | 48w03 | (44·10n-773)/9 | {3,7,13,37} | 6 | 3,37,13,3,37,7 |
| 37111 | 371w | (334·10n-1)/9 | {3,7,13,37} | 6 | 37,7,3,37,13,3 |
| - | 271469w | 27147·10n-1 | {7,11,13,37} | 6 | 7,11,13,11,37,11 |
| - | 93510w1 | 9351·10n+1 | {7,11,13,37} | 6 | 7,11,37,11,13,11 |
| - | 360479w | 36048·10n-1 | {7,11,13,37} | 6 | 11,7,11,13,11,37 |
| - | 101759w | 10176·10n-1 | {7,11,13,37} | 6 | 11,7,11,37,11,13 |
| - | 176760w1 | 17676·10n+1 | {7,11,13,37} | 6 | 11,13,11,7,11,37 |
| - | 248260w1 | 24826·10n+1 | {7,11,13,37} | 6 | 11,13,11,37,11,7 |
| - | 359159w | 35916·10n-1 | {7,11,13,37} | 6 | 11,37,11,7,11,13 |
| - | 176009w | 17601·10n-1 | {7,11,13,37} | 6 | 11,37,11,13,11,7 |
| - | 286120w1 | 28612·10n+1 | {7,11,13,37} | 6 | 13,11,7,11,37,11 |
| - | 260380w1 | 26038·10n+1 | {7,11,13,37} | 6 | 13,11,37,11,7,11 |
| - | 258269w | 25827·10n-1 | {7,11,13,37} | 6 | 37,11,7,11,13,11 |
| - | 91750w1 | 9175·10n+1 | {7,11,13,37} | 6 | 37,11,13,11,7,11 |
| - | 44596649w | 4459665·10n-1 | {11,73,101,137} | 8 | 11,73,11,101,11,137,11,101 |
| - | 94890409w | 9489041·10n-1 | {11,73,101,137} | 8 | 11,101,11,73,11,101,11,137 |
| - | 95890510w1 | 9589051·10n+1 | {11,73,101,137} | 8 | 11,101,11,137,11,101,11,73 |
| - | 33495540w1 | 3349554·10n+1 | {11,73,101,137} | 8 | 11,137,11,101,11,73,11,101 |
| - | 109589050w1 | 10958905·10n+1 | {11,73,101,137} | 8 | 73,11,101,11,137,11,101,11 |
| - | 51095890w1 | 5109589·10n+1 | {11,73,101,137} | 8 | 101,11,73,11,101,11,137,11 |
| - | 41094889w | 4109489·10n-1 | {11,73,101,137} | 8 | 101,11,137,11,101,11,73,11 |
| - | 1622070w1 | 162207·10n+1 | {11,73,101,137} | 8 | 137,11,101,11,73,11,101,11 |
| - | 1313368639w | 131336864·10n-1 | {7,11,13,101,9901} | 12 | 7,11,101,11,13,11,101,11,9901,11,101,11 |
| - | 5936739610w1 | 593673961·10n+1 | {7,11,13,101,9901} | 12 | 7,11,101,11,9901,11,101,11,13,11,101,11 |
| - | 6137342869w | 613734287·10n-1 | {7,11,13,101,9901} | 12 | 11,7,11,101,11,13,11,101,11,9901,11,101 |
| - | 407327039w | 40732704·10n-1 | {7,11,13,101,9901} | 12 | 11,7,11,101,11,9901,11,101,11,13,11,101 |
| - | 1297338029w | 129733803·10n-1 | {7,11,13,101,9901} | 12 | 11,13,11,101,11,7,11,101,11,9901,11,101 |
| - | 2057328689w | 205732869·10n-1 | {7,11,13,101,9901} | 12 | 11,13,11,101,11,9901,11,101,11,7,11,101 |
| - | 1362674729w | 136267473·10n-1 | {7,11,13,101,9901} | 12 | 11,101,11,7,11,101,11,13,11,101,11,9901 |
| - | 396327130w1 | 39632713·10n+1 | {7,11,13,101,9901} | 12 | 11,101,11,7,11,101,11,9901,11,101,11,13 |
| - | 287326920w1 | 28732692·10n+1 | {7,11,13,101,9901} | 12 | 11,101,11,13,11,101,11,7,11,101,11,9901 |
| - | 6886333620w1 | 688633362·10n+1 | {7,11,13,101,9901} | 12 | 11,101,11,13,11,101,11,9901,11,101,11,7 |
| - | 4477341210w1 | 447734121·10n+1 | {7,11,13,101,9901} | 12 | 11,101,11,9901,11,101,11,7,11,101,11,13 |
| - | 692663959w | 69266396·10n-1 | {7,11,13,101,9901} | 12 | 11,101,11,9901,11,101,11,13,11,101,11,7 |
| - | 2716329449w | 271632945·10n-1 | {7,11,13,101,9901} | 12 | 11,9901,11,101,11,7,11,101,11,13,11,101 |
| - | 10813684080w1 | 1081368408·10n+1 | {7,11,13,101,9901} | 12 | 11,9901,11,101,11,13,11,101,11,7,11,101 |
| - | 2963370289w | 296337029·10n-1 | {7,11,13,101,9901} | 12 | 13,11,101,11,7,11,101,11,9901,11,101,11 |
| - | 9456743130w1 | 945674313·10n+1 | {7,11,13,101,9901} | 12 | 13,11,101,11,9901,11,101,11,7,11,101,11 |
| - | 3616737289w | 361673729·10n-1 | {7,11,13,101,9901} | 12 | 101,11,7,11,101,11,13,11,101,11,9901,11 |
| - | 6046738709w | 604673871·10n-1 | {7,11,13,101,9901} | 12 | 101,11,7,11,101,11,9901,11,101,11,13,11 |
| - | 12883279210w1 | 1288327921·10n+1 | {7,11,13,101,9901} | 12 | 101,11,13,11,101,11,7,11,101,11,9901,11 |
| - | 8803276140w1 | 880327614·10n+1 | {7,11,13,101,9901} | 12 | 101,11,13,11,101,11,9901,11,101,11,7,11 |
| - | 5276637949w | 527663795·10n-1 | {7,11,13,101,9901} | 12 | 101,11,9901,11,101,11,7,11,101,11,13,11 |
| - | 13093380420w1 | 1309338042·10n+1 | {7,11,13,101,9901} | 12 | 101,11,9901,11,101,11,13,11,101,11,7,11 |
| - | 2866735530w1 | 286673553·10n+1 | {7,11,13,101,9901} | 12 | 9901,11,101,11,7,11,101,11,13,11,101,11 |
| - | 1973269309w | 197326931·10n-1 | {7,11,13,101,9901} | 12 | 9901,11,101,11,13,11,101,11,7,11,101,11 |
| - | 993267319w | 99326732·10n-1 | {7,11,37,101,9901} | 12 | 7,11,37,11,9901,11,101,11,37,11,101,11 |
| - | 9466642140w1 | 946664214·10n+1 | {7,11,37,101,9901} | 12 | 7,11,101,11,37,11,101,11,9901,11,37,11 |
| - | 22692696060w1 | 2269269606·10n+1 | {7,11,37,101,9901} | 12 | 11,7,11,37,11,9901,11,101,11,37,11,101 |
| - | 12342675610w1 | 1234267561·10n+1 | {7,11,37,101,9901} | 12 | 11,7,11,101,11,37,11,101,11,9901,11,37 |
| - | 3257329890w1 | 325732989·10n+1 | {7,11,37,101,9901} | 12 | 11,37,11,7,11,101,11,37,11,101,11,9901 |
| - | 23627360359w | 2362736036·10n-1 | {7,11,37,101,9901} | 12 | 11,37,11,101,11,7,11,37,11,9901,11,101 |
| - | 7952681320w1 | 795268132·10n+1 | {7,11,37,101,9901} | 12 | 11,37,11,101,11,9901,11,37,11,7,11,101 |
| - | 38422701689w | 3842270169·10n-1 | {7,11,37,101,9901} | 12 | 11,37,11,9901,11,101,11,37,11,101,11,7 |
| - | 1936328670w1 | 193632867·10n+1 | {7,11,37,101,9901} | 12 | 11,101,11,7,11,37,11,9901,11,101,11,37 |
| - | 38982712349w | 3898271235·10n-1 | {7,11,37,101,9901} | 12 | 11,101,11,37,11,101,11,7,11,37,11,9901 |
| - | 9196335930w1 | 919633593·10n+1 | {7,11,37,101,9901} | 12 | 11,101,11,37,11,101,11,9901,11,37,11,7 |
| - | 2452665719w | 245266572·10n-1 | {7,11,37,101,9901} | 12 | 11,101,11,9901,11,37,11,7,11,101,11,37 |
| - | 11143684410w1 | 1114368441·10n+1 | {7,11,37,101,9901} | 12 | 11,9901,11,37,11,7,11,101,11,37,11,101 |
| - | 3883677150w1 | 388367715·10n+1 | {7,11,37,101,9901} | 12 | 11,9901,11,101,11,37,11,101,11,7,11,37 |
| - | 24406758079w | 2440675808·10n-1 | {7,11,37,101,9901} | 12 | 37,11,7,11,101,11,37,11,101,11,9901,11 |
| - | 8353375679w | 835337568·10n-1 | {7,11,37,101,9901} | 12 | 37,11,101,11,7,11,37,11,9901,11,101,11 |
| - | 5943272269w | 594327227·10n-1 | {7,11,37,101,9901} | 12 | 37,11,101,11,9901,11,37,11,7,11,101,11 |
| - | 13856646529w | 1385664653·10n-1 | {7,11,37,101,9901} | 12 | 37,11,9901,11,101,11,37,11,101,11,7,11 |
| - | 37616770279w | 3761677028·10n-1 | {7,11,37,101,9901} | 12 | 101,11,7,11,37,11,9901,11,101,11,37,11 |
| - | 19456753129w | 1945675313·10n-1 | {7,11,37,101,9901} | 12 | 101,11,37,11,101,11,7,11,37,11,9901,11 |
| - | 6493273830w1 | 649327383·10n+1 | {7,11,37,101,9901} | 12 | 101,11,37,11,101,11,9901,11,37,11,7,11 |
| - | 3963371290w1 | 396337129·10n+1 | {7,11,37,101,9901} | 12 | 101,11,9901,11,37,11,7,11,101,11,37,11 |
| - | 25966758630w1 | 2596675863·10n+1 | {7,11,37,101,9901} | 12 | 9901,11,37,11,7,11,101,11,37,11,101,11 |
| - | 18143285479w | 1814328548·10n-1 | {7,11,37,101,9901} | 12 | 9901,11,101,11,37,11,101,11,7,11,37,11 |
| - | 22077348709w | 2207734871·10n-1 | {11,13,37,101,9901} | 12 | 11,13,11,37,11,9901,11,101,11,37,11,101 |
| - | 52777389509w | 5277738951·10n-1 | {11,13,37,101,9901} | 12 | 11,13,11,101,11,37,11,101,11,9901,11,37 |
| - | 21162694529w | 2116269453·10n-1 | {11,13,37,101,9901} | 12 | 11,37,11,13,11,101,11,37,11,101,11,9901 |
| - | 3277340009w | 327734001·10n-1 | {11,13,37,101,9901} | 12 | 11,37,11,101,11,13,11,37,11,9901,11,101 |
| - | 3882677250w1 | 388267725·10n+1 | {11,13,37,101,9901} | 12 | 11,37,11,101,11,9901,11,37,11,13,11,101 |
| - | 14487351220w1 | 1448735122·10n+1 | {11,13,37,101,9901} | 12 | 11,37,11,9901,11,101,11,37,11,101,11,13 |
| - | 42636369370w1 | 4263636937·10n+1 | {11,13,37,101,9901} | 12 | 11,101,11,13,11,37,11,9901,11,101,11,37 |
| - | 47122720489w | 4712272049·10n-1 | {11,13,37,101,9901} | 12 | 11,101,11,37,11,101,11,13,11,37,11,9901 |
| - | 39643712909w | 3964371291·10n-1 | {11,13,37,101,9901} | 12 | 11,101,11,37,11,101,11,9901,11,37,11,13 |
| - | 35012698279w | 3501269828·10n-1 | {11,13,37,101,9901} | 12 | 11,101,11,9901,11,37,11,13,11,101,11,37 |
| - | 9206335939w | 920633594·10n-1 | {11,13,37,101,9901} | 12 | 11,9901,11,37,11,13,11,101,11,37,11,101 |
| - | 32746359479w | 3274635948·10n-1 | {11,13,37,101,9901} | 12 | 11,9901,11,101,11,37,11,101,11,13,11,37 |
| - | 43776777450w1 | 4377677745·10n+1 | {11,13,37,101,9901} | 12 | 13,11,37,11,9901,11,101,11,37,11,101,11 |
| - | 51583418909w | 5158341891·10n-1 | {11,13,37,101,9901} | 12 | 13,11,101,11,37,11,101,11,9901,11,37,11 |
| - | 13266340w1 | 1326634·10n+1 | {11,13,37,101,9901} | 12 | 37,11,13,11,101,11,37,11,101,11,9901,11 |
| - | 66993333319w | 6699333332·10n-1 | {11,13,37,101,9901} | 12 | 37,11,101,11,9901,11,37,11,13,11,101,11 |
| - | 66766699439w | 6676669944·10n-1 | {11,13,37,101,9901} | 12 | 37,11,9901,11,101,11,37,11,101,11,13,11 |
| - | 49826782489w | 4982678249·10n-1 | {11,13,37,101,9901} | 12 | 101,11,13,11,37,11,9901,11,101,11,37,11 |
| - | 4963271290w1 | 496327129·10n+1 | {11,13,37,101,9901} | 12 | 101,11,37,11,101,11,13,11,37,11,9901,11 |
| - | 26066758729w | 2606675873·10n-1 | {11,13,37,101,9901} | 12 | 101,11,37,11,101,11,9901,11,37,11,13,11 |
| - | 32666665339w | 3266666534·10n-1 | {11,13,37,101,9901} | 12 | 101,11,9901,11,37,11,13,11,101,11,37,11 |
| - | 66666799330w1 | 6666679933·10n+1 | {11,13,37,101,9901} | 12 | 9901,11,37,11,13,11,101,11,37,11,101,11 |
| - | 10003277339w | 1000327734·10n-1 | {11,13,37,101,9901} | 12 | 9901,11,101,11,37,11,101,11,13,11,37,11 |
| - | 14774658240w1 | 1477465824·10n+1 | {7,11,13,73,137,9901} | 24 | 11,7,11,137,11,13,11,7,11,9901,11,13, 11,7,11,73,11,13,11,7,11,9901,11,13 |
| - | 8109503059w | 810950306·10n-1 | {7,11,13,73,137,9901} | 24 | 11,7,11,9901,11,13,11,7,11,73,11,13, 11,7,11,9901,11,13,11,7,11,137,11,13 |
| - | 59390257510w1 | 5939025751·10n+1 | {7,11,13,73,137,9901} | 24 | 11,13,11,7,11,9901,11,13,11,7,11,73, 11,13,11,7,11,9901,11,13,11,7,11,137 |
| - | 66054412590w1 | 6605441259·10n+1 | {7,11,13,73,137,9901} | 24 | 11,73,11,13,11,7,11,9901,11,13,11,7, 11,137,11,13,11,7,11,9901,11,13,11,7 |
| - | 49611238720w1 | 4961123872·10n+1 | {7,11,37,73,137,9901} | 24 | 37,11,9901,11,7,11,37,11,73,11,7,11, 37,11,9901,11,7,11,37,11,137,11,7,11 |
| - | 39400930480w1 | 3940093048·10n+1 | {7,11,37,73,137,9901} | 24 | 9901,11,7,11,37,11,137,11,7,11,37,11, 9901,11,7,11,37,11,73,11,7,11,37,11 |
| - | 6208139830w1 | 620813983·10n+1 | {7,13,37,73,101,137} | 24 | 7,37,13,73,37,101,7,37,13,101,37,73, 7,37,13,137,37,101,7,37,13,101,37,137 |
| - | 4073899540w1 | 407389954·10n+1 | {7,13,37,73,101,137} | 24 | 7,37,13,101,37,73,7,37,13,137,37,101, 7,37,13,101,37,137,7,37,13,73,37,101 |
| - | 18438535859w | 1843853586·10n-1 | {7,13,37,73,101,137} | 24 | 7,37,13,101,37,137,7,37,13,73,37,101, 7,37,13,101,37,73,7,37,13,137,37,101 |
| - | 368755949w | 36875595·10n-1 | {7,13,37,73,101,137} | 24 | 7,37,13,137,37,101,7,37,13,101,37,137, 7,37,13,73,37,101,7,37,13,101,37,73 |
| - | 23882980249w | 2388298025·10n-1 | {7,13,37,73,101,137} | 24 | 7,73,37,101,13,37,7,101,37,73,13,37, 7,137,37,101,13,37,7,101,37,137,13,37 |
| - | 33246405229w | 3324640523·10n-1 | {7,13,37,73,101,137} | 24 | 7,101,37,73,13,37,7,137,37,101,13,37, 7,101,37,137,13,37,7,73,37,101,13,37 |
| - | 5813200339w | 581320034·10n-1 | {7,13,37,73,101,137} | 24 | 7,101,37,137,13,37,7,73,37,101,13,37, 7,101,37,73,13,37,7,137,37,101,13,37 |
| - | 32639555820w1 | 3263955582·10n+1 | {7,13,37,73,101,137} | 24 | 7,137,37,101,13,37,7,101,37,137,13,37, 7,73,37,101,13,37,7,101,37,73,13,37 |
| - | 8741354969w | 874135497·10n-1 | {7,13,37,73,101,137} | 24 | 13,37,7,73,37,101,13,37,7,101,37,73, 13,37,7,137,37,101,13,37,7,101,37,137 |
| - | 10875595259w | 1087559526·10n-1 | {7,13,37,73,101,137} | 24 | 13,37,7,101,37,73,13,37,7,137,37,101, 13,37,7,101,37,137,13,37,7,73,37,101 |
| - | 34632170680w1 | 3463217068·10n+1 | {7,13,37,73,101,137} | 24 | 13,37,7,101,37,137,13,37,7,73,37,101, 13,37,7,101,37,73,13,37,7,137,37,101 |
| - | 18691849920w1 | 1869184992·10n+1 | {7,13,37,73,101,137} | 24 | 13,37,7,137,37,101,13,37,7,101,37,137, 13,37,7,73,37,101,13,37,7,101,37,73 |
| - | 8632170940w1 | 863217094·10n+1 | {7,13,37,73,101,137} | 24 | 13,73,37,101,7,37,13,101,37,73,7,37, 13,137,37,101,7,37,13,101,37,137,7,37 |
| - | 731254039w | 73125404·10n-1 | {7,13,37,73,101,137} | 24 | 13,101,37,73,7,37,13,137,37,101,7,37, 13,101,37,137,7,37,13,73,37,101,7,37 |
| - | 7308149819w | 730814982·10n-1 | {7,13,37,73,101,137} | 24 | 13,101,37,137,7,37,13,73,37,101,7,37, 13,101,37,73,7,37,13,137,37,101,7,37 |
| - | 2865494329w | 286549433·10n-1 | {7,13,37,73,101,137} | 24 | 13,137,37,101,7,37,13,101,37,137,7,37, 13,73,37,101,7,37,13,101,37,73,7,37 |
| - | 14616752310w1 | 1461675231·10n+1 | {7,13,37,73,101,137} | 24 | 37,7,73,37,101,13,37,7,101,37,73,13, 37,7,137,37,101,13,37,7,101,37,137,13 |
| - | 27284450080w1 | 2728445008·10n+1 | {7,13,37,73,101,137} | 24 | 37,7,101,37,73,13,37,7,137,37,101,13, 37,7,101,37,137,13,37,7,73,37,101,13 |
| - | 40230800770w1 | 4023080077·10n+1 | {7,13,37,73,101,137} | 24 | 37,7,101,37,137,13,37,7,73,37,101,13, 37,7,101,37,73,13,37,7,137,37,101,13 |
| - | 16867995850w1 | 1686799585·10n+1 | {7,13,37,73,101,137} | 24 | 37,7,137,37,101,13,37,7,101,37,137,13, 37,7,73,37,101,13,37,7,101,37,73,13 |
| - | 28071297630w1 | 2807129763·10n+1 | {7,13,37,73,101,137} | 24 | 37,13,73,37,101,7,37,13,101,37,73,7, 37,13,137,37,101,7,37,13,101,37,137,7 |
| - | 6728894730w1 | 672889473·10n+1 | {7,13,37,73,101,137} | 24 | 37,13,101,37,73,7,37,13,137,37,101,7, 37,13,101,37,137,7,37,13,73,37,101,7 |
| - | 19675245420w1 | 1967524542·10n+1 | {7,13,37,73,101,137} | 24 | 37,13,101,37,137,7,37,13,73,37,101,7, 37,13,101,37,73,7,37,13,137,37,101,7 |
| - | 30322541170w1 | 3032254117·10n+1 | {7,13,37,73,101,137} | 24 | 37,13,137,37,101,7,37,13,101,37,137,7, 37,13,73,37,101,7,37,13,101,37,73,7 |
| - | 5105202659w | 510520266·10n-1 | {7,13,37,73,101,137} | 24 | 37,73,7,37,13,137,37,101,7,37,13,101, 37,137,7,37,13,73,37,101,7,37,13,101 |
| - | 42359443330w1 | 4235944333·10n+1 | {7,13,37,73,101,137} | 24 | 37,73,13,37,7,137,37,101,13,37,7,101, 37,137,13,37,7,73,37,101,13,37,7,101 |
| - | 12964577250w1 | 1296457725·10n+1 | {7,13,37,73,101,137} | 24 | 37,101,7,37,13,101,37,73,7,37,13,137, 37,101,7,37,13,101,37,137,7,37,13,73 |
| - | 14468627639w | 1446862764·10n-1 | {7,13,37,73,101,137} | 24 | 37,101,7,37,13,101,37,137,7,37,13,73, 37,101,7,37,13,101,37,73,7,37,13,137 |
| - | 26419122570w1 | 2641912257·10n+1 | {7,13,37,73,101,137} | 24 | 37,101,13,37,7,101,37,73,13,37,7,137, 37,101,13,37,7,101,37,137,13,37,7,73 |
| - | 1014082319w | 101408232·10n-1 | {7,13,37,73,101,137} | 24 | 37,101,13,37,7,101,37,137,13,37,7,73, 37,101,13,37,7,101,37,73,13,37,7,137 |
| - | 16602867929w | 1660286793·10n-1 | {7,13,37,73,101,137} | 24 | 37,137,7,37,13,73,37,101,7,37,13,101, 37,73,7,37,13,137,37,101,7,37,13,101 |
| - | 3148322609w | 314832261·10n-1 | {7,13,37,73,101,137} | 24 | 37,137,13,37,7,73,37,101,13,37,7,101, 37,73,13,37,7,137,37,101,13,37,7,101 |
| - | 17041925929w | 1704192593·10n-1 | {7,13,37,73,101,137} | 24 | 73,7,37,13,137,37,101,7,37,13,101,37, 137,7,37,13,73,37,101,7,37,13,101,37 |
| - | 15473225260w1 | 1547322526·10n+1 | {7,13,37,73,101,137} | 24 | 73,13,37,7,137,37,101,13,37,7,101,37, 137,13,37,7,73,37,101,13,37,7,101,37 |
| - | 15708592609w | 1570859261·10n-1 | {7,13,37,73,101,137} | 24 | 73,37,101,7,37,13,101,37,73,7,37,13, 137,37,101,7,37,13,101,37,137,7,37,13 |
| - | 33251002860w1 | 3325100286·10n+1 | {7,13,37,73,101,137} | 24 | 73,37,101,13,37,7,101,37,73,13,37,7, 137,37,101,13,37,7,101,37,137,13,37,7 |
| - | 27615470490w1 | 2761547049·10n+1 | {7,13,37,73,101,137} | 24 | 101,7,37,13,101,37,73,7,37,13,137,37, 101,7,37,13,101,37,137,7,37,13,73,37 |
| - | 8645873719w | 864587372·10n-1 | {7,13,37,73,101,137} | 24 | 101,7,37,13,101,37,137,7,37,13,73,37, 101,7,37,13,101,37,73,7,37,13,137,37 |
| - | 7889579659w | 788957966·10n-1 | {7,13,37,73,101,137} | 24 | 101,13,37,7,101,37,73,13,37,7,137,37, 101,13,37,7,101,37,137,13,37,7,73,37 |
| - | 23869277470w1 | 2386927747·10n+1 | {7,13,37,73,101,137} | 24 | 101,13,37,7,101,37,137,13,37,7,73,37, 101,13,37,7,101,37,73,13,37,7,137,37 |
| - | 41322641069w | 4132264107·10n-1 | {7,13,37,73,101,137} | 24 | 101,37,73,7,37,13,137,37,101,7,37,13, 101,37,137,7,37,13,73,37,101,7,37,13 |
| - | 26373146269w | 2637314627·10n-1 | {7,13,37,73,101,137} | 24 | 101,37,73,13,37,7,137,37,101,13,37,7, 101,37,137,13,37,7,73,37,101,13,37,7 |
| - | 28948803810w1 | 2894880381·10n+1 | {7,13,37,73,101,137} | 24 | 101,37,137,7,37,13,73,37,101,7,37,13, 101,37,73,7,37,13,137,37,101,7,37,13 |
| - | 9888197940w1 | 988819794·10n+1 | {7,13,37,73,101,137} | 24 | 101,37,137,13,37,7,73,37,101,13,37,7, 101,37,73,13,37,7,137,37,101,13,37,7 |
| - | 29988276619w | 2998827662·10n-1 | {7,13,37,73,101,137} | 24 | 137,7,37,13,73,37,101,7,37,13,101,37, 73,7,37,13,137,37,101,7,37,13,101,37 |
| - | 2526874570w1 | 252687457·10n+1 | {7,13,37,73,101,137} | 24 | 137,13,37,7,73,37,101,13,37,7,101,37, 73,13,37,7,137,37,101,13,37,7,101,37 |
| - | 39365258040w1 | 3936525804·10n+1 | {7,13,37,73,101,137} | 24 | 137,37,101,7,37,13,101,37,137,7,37,13, 73,37,101,7,37,13,101,37,73,7,37,13 |
| - | 20304652170w1 | 2030465217·10n+1 | {7,13,37,73,101,137} | 24 | 137,37,101,13,37,7,101,37,137,13,37,7, 73,37,101,13,37,7,101,37,73,13,37,7 |
| - | 40886080390w1 | 4088608039·10n+1 | {11,13,37,73,137,9901} | 24 | 11,13,11,37,11,73,11,13,11,37,11,9901, 11,13,11,37,11,137,11,13,11,37,11,9901 |
| - | 35669710349w | 3566971035·10n-1 | {11,17,101,137,353,641} | 32 | 11,101,11,137,11,101,11,353,11,101,11,137, 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 | |||||
| - | 17187514510w1 | 1718751451·10n+1 | {7,11,13,31,41,61,101,211,241} | 60 | 11,101,11,13,11,101,11,241,11,101,11,41, 11,101,11,13,11,101,11,61,11,101,11,7, 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 | |||||
| - | 54466351710w1 | 5446635171·10n+1 | {11,13,17,37,73,97,101,641} | 96 | 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, 11,97,11,37,11,13,11,101,11,37,11,101, 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 | |||||
| - | 25283561389w | 2528356139·10n-1 | {11,13,17,37,97,101,137,449} | 96 | 11,37,11,101,11,13,11,37,11,97,11,101, 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, 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 | |||||