Covering set of near-repdigit-related sequences

Table of contents目次

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 $91 w 3 = {93, 913, 9113, 91113, 911113, \dots} = (82 \times 10^{n} + 17) / 9$ 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 $91 w 3 = {93, 913, 9113, 91113, 911113, \dots} = (82 \times 10^{n} + 17) / 9$ は被覆集合 ${3, 7, 11, 13}$ で完全に覆われます。

\begin{aligned} (82 \times 10^{3 k + 1} + 17) / 9 & = 3 \times [31 + 30340 \times (10^{3 k} - 1) / (10^{3} - 1)] \\ = 3 \times [31 + 30340 \times \sum_{m = 0}^{k - 1} 10^{3 m}] \\ (82 \times 10^{6 k + 5} + 17) / 9 & = 7 \times [130159 + 130158600000 \times (10^{6 k} - 1) / (10^{6} - 1)] \\ = 7 \times [130159 + 130158600000 \times \sum_{m = 0}^{k - 1} 10^{6 m}] \\ (82 \times 10^{2 k} + 17) / 9 & = 11 \times [1 + 82 \times (10^{2 k} - 1) / (10^{2} - 1)] \\ = 11 \times [1 + 82 \times \sum_{m = 0}^{k - 1} 10^{2 m}] \\ (82 \times 10^{6 k + 3} + 17) / 9 & = 13 \times [701 + 700854000 \times (10^{6 k} - 1) / (10^{6} - 1)] \\ = 13 \times [701 + 700854000 \times \sum_{m = 0}^{k - 1} 10^{6 m}] \end{aligned}

\begin{array}{ccccccc} n & 6 k & 6 k + 1 & 6 k + 2 & 6 k + 3 & 6 k + 4 & 6 k + 5 \\ 3 & \cdot & 3 k + 1 & \cdot & \cdot & 3 k + 1 & \cdot \\ 7 & \cdot & \cdot & \cdot & \cdot & \cdot & 6 k + 5 \\ 11 & 2 k & \cdot & 2 k & \cdot & 2 k & \cdot \\ 13 & \cdot & \cdot & \cdot & 6 k + 3 & \cdot & \cdot \\ (82 \times 10^{n} + 17) / 9 is divisible by & 11 & 3 & 11 & 13 & 3, 11 & 7 \end{array}

2.2. 381w

Quasi-repdigit sequence $381 w = {38, 381, 3811, 38111, 381111, \dots} = (343 \times 10^{n} - 1) / 9$ 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 $381 w = {38, 381, 3811, 38111, 381111, \dots} = (343 \times 10^{n} - 1) / 9$ は被覆集合 ${3, 37, 23 w}$ で完全に覆われます。

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

\begin{aligned} (343 \times 10^{3 k + 1} - 1) / 9 & = 3 \times [127 + 126910 \times (10^{3 k} - 1) / (10^{3} - 1)] \\ = 3 \times [127 + 126910 \times \sum_{m = 0}^{k - 1} 10^{3 m}] \\ (343 \times 10^{3 k + 2} - 1) / 9 & = 37 \times [103 + 102900 \times (10^{3 k} - 1) / (10^{3} - 1)] \\ = 37 \times [103 + 102900 \times \sum_{m = 0}^{k - 1} 10^{3 m}] \\ (343 \times 10^{3 k} - 1) / 9 & = (7 \times 10^{k} - 1) / 3 \times (49 \times 10^{2 k} + 7 \times 10^{k} + 1) / 3 \\ = 23 w \times 163 w 56 w 7 \end{aligned}

\begin{array}{cccc} n & 3 k & 3 k + 1 & 3 k + 2 \\ 3 & \cdot & 3 k + 1 & \cdot \\ 37 & \cdot & \cdot & 3 k + 2 \\ 23 w & 3 k & \cdot & \cdot \\ (343 \times 10^{n} - 1) / 9 is divisible by & 23 w & 3 & 37 \end{array}

2.3. 13266340w1

Near-repdigit-related sequence $13266340 w 1 = {13266341, 132663401, 1326634001, 13266340001, 132663400001, \dots} = 1326634 \times 10^{n} + 1$ 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 $13266340 w 1 = {13266341, 132663401, 1326634001, 13266340001, 132663400001, \dots} = 1326634 \times 10^{n} + 1$ は被覆集合 ${11, 13, 37, 101, 9901}$ で完全に覆われます。

\begin{aligned} 1326634 \times 10^{2 k + 1} + 1 & = 11 \times [1206031 + 119397060 \times (10^{2 k} - 1) / (10^{2} - 1)] \\ = 11 \times [1206031 + 119397060 \times \sum_{m = 0}^{k - 1} 10^{2 m}] \\ 1326634 \times 10^{6 k + 2} + 1 & = 13 \times [10204877 + 10204866718200 \times (10^{6 k} - 1) / (10^{6} - 1)] \\ = 13 \times [10204877 + 10204866718200 \times \sum_{m = 0}^{k - 1} 10^{6 m}] \\ 1326634 \times 10^{3 k} + 1 & = 37 \times [35855 + 35819118 \times (10^{3 k} - 1) / (10^{3} - 1)] \\ = 37 \times [35855 + 35819118 \times \sum_{m = 0}^{k - 1} 10^{3 m}] \\ 1326634 \times 10^{4 k} + 1 & = 101 \times [13135 + 131336766 \times (10^{4 k} - 1) / (10^{4} - 1)] \\ = 101 \times [13135 + 131336766 \times \sum_{m = 0}^{k - 1} 10^{4 m}] \\ 1326634 \times 10^{12 k + 10} + 1 & = 9901 \times [1339899000101 + 1339899000099660000000000 \times (10^{12 k} - 1) / (10^{12} - 1)] \\ = 9901 \times [1339899000101 + 1339899000099660000000000 \times \sum_{m = 0}^{k - 1} 10^{12 m}] \end{aligned}

\begin{array}{ccccccc} n & 12 k & 12 k + 1 & 12 k + 2 & 12 k + 3 & 12 k + 4 & 12 k + 5 & 12 k + 6 & 12 k + 7 & 12 k + 8 & 12 k + 9 & 12 k + 10 & 12 k + 11 \\ 11 & \cdot & 2 k + 1 & \cdot & 2 k + 1 & \cdot & 2 k + 1 & \cdot & 2 k + 1 & \cdot & 2 k + 1 & \cdot & 2 k + 1 \\ 13 & \cdot & \cdot & 6 k + 2 & \cdot & \cdot & \cdot & \cdot & \cdot & 6 k + 2 & \cdot & \cdot & \cdot \\ 37 & 3 k & \cdot & \cdot & 3 k & \cdot & \cdot & 3 k & \cdot & \cdot & 3 k & \cdot & \cdot \\ 101 & 4 k & \cdot & \cdot & \cdot & 4 k & \cdot & \cdot & \cdot & 4 k & \cdot & \cdot & \cdot \\ 9901 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 12 k + 10 & \cdot \\ 1326634 \times 10^{n} + 1 is divisible by & 37, 101 & 11 & 13 & 11, 37 & 101 & 11 & 37 & 11 & 13, 101 & 11, 37 & 9901 & 11 \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
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				32
-	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				60
-	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				96
-	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				96