Composite Sequences

In the February 2011 Mathematical Monthly Lenny Jones [1] considered the following question:
For a given digit d determine the smallest value k relatively prime to d so that when the digit d is repeatedly appended to the end of k the sequence kd, kdd, kddd, ... contains only composite numbers.
Jones proved that for d=1 the smallest k is 37. and noted that for d=3, k <= 4070, d=7, k <= 606474, and d=9, k <= 1879711.
Grantham, Jarnicki, Rickert and Wagon (GJRW, to appear) determined that for d=3, k = 4070 or 817, d=7, k = 891, d=9, k = 10175 or 8578 or 7018 or 4420 and conjectured that k₃ = 4070, k₉ = 10175.
In the same paper GJRW studied the question in other bases and determined answers for several cases and conjectures for most of the bases less than ten.
For each base b and trailing digit d relatively prime to the base this table lists the smallest known seed k so that appending the digit d any positive integer number of times to the base b representation of k always produces a composite number. The first column is the base b, the second column is the digit appended d, the third column is the base ten representation of the smallest known seed k, the fourth column is the base b representation of the smallest known seed, the fifth column is the number of values relatively prime to d that are less than k for which no prime is yet known for the sequence.

Base	Digit	seed base 10	seed base b	Candidates
2	1	509202	1111100010100010010(1)	58	H. Riesel
3	1	6059	22022102(1)	15
	2	63064644937	20000210002020220021021(2)		CRUS
4	1	5	11(1)	Minimum
	3	8	20(3)	Minimum
5	1	3	3(1)	Minimum
	2	191115	22103430(2)	258
	3	585655	122220110(3)	2274
	4	346801	42044201(4)	126
6	1	26789	324005(1)	29
	5	84686	1452022(5)	2
7	1	76	136(1)	Minimum*
	2	15979	64405(2)	17
	3	5629	22261(3)	19
	4	20277	113055(4)	16
	5	43	2211(5)	Minimum
	6	408034255081	41323316641135(6)		CRUS
8	1	21	25(1)	Minimum
	3	1079770	4074732(3)	721
	5	7476	16464(5)	29
	7	13	15(7)	Minimum
9	1	1	1(1)	Minimum
	2	4615	6287(2)	14
	4	6059	8272(4)	12
	5	78	86(5)	Minimum*
	7	2	2(7)	Minimum
	8	3	3(8)	Minimum
10	1	37	37(1)	Minimum	L. Jones
	3	4070	4070(3)	1	L. Jones, S. Wagon
	7	891	891(7)	Minimum*	S. Wagon
	9	10175	10175(9)	3	CRUS

* In these cases some of the primes found for smaller seeds are probable primes that have not been certified.
H. Riesel, Några stora primtal (Swedish: Some large primes), Elementa 39 (1956), 258-260.
L. Jones, When does appending the same digit repeatedly on the right of a positive integer generate a sequence of composite numbers?, Amer. Math. Monthly 118 (2011), 153-160.

Base 3

Digit 1

The prime factors of 6059 follow a pattern of period 12 using the cover

Residue	1	2	2	4	0
Modulus	2	3	4	6	12
Prime	2	13	5	7	73

The primes dividing 12A+r are

0	1	2	3	4	5	6	7	8	9	10	11
73	2	5	2	7	2	5	2	13	2	5	2

In list form {73,2,5,2,7,2,5,2,13,2,5,2}.
Candidates checked to 19200 digits.
(806,2419), (915,2746), 2877, (968,2905), 3059, 3393, 3689, 3738, 3755, 3813, 3969, 4029, (156, 469, 1408, 4225), 4356, 4388, (498, 1495, 4486), 4589, (1565,4696), 4905, 5325, (1794,5383), 5625, 5798, 5876, 6059.
Values with first prime occurring with more than 5000 digits.

k	exponent	k	exponent	k	exponent
2055	12978,	2718	9567,	3059	28580
3158	15331	3368	17455	3515	5898
3755	26022	3788	5031	4225	24758
4376	16533	4486	20845	4589	21404
4655	11134	4819	8968	5005	12082
5625	24314	5759	11140

The remaining 17 values, and how far they've been checked. There are 15 candidates remaining through 50000 digits.

value	prime at	composite to	value	prime at	composite to
(806,2419)		50000	4029	47256
(915,2746)		50000	4356		50000
(968,2905)		50000	4388		50000
(1565,4696)		50000	4905		50000
2877		50000	5325		50000
3393		50000	5383		50000
3738		50000	5798		50000
3813		50000	5876	36665
3969		50000

Digit 2

56532669941364262542767 has period 120 using the cover

0	3	4	2	1	2	5	13	17	13	16	10
3	4	5	6	8	10	12	15	20	24	30	30
13	5	11	7	41	61	73	4561	1181	6481	31	271

The primes dividing 120A + r are

r	0	1	2	3	4	5	6	7	8	9
prime	13	41	7	13	11	73	13	5	7	13
r	10	11	12	13	14	15	16	17	18	19
prime	271	5	13	4561	11	13	31	41	13	5
r	20	21	22	23	24	25	26	27	28	29
prime	7	13	61	5	13	41	7	13	4561	11
r	30	31	32	33	34	35	36	37	38	39
prime	13	5	7	13	11	5	13	1181	7	13
r	40	41	42	43	44	45	46	47	48	49
prime	271	41	13	5	11	13	31	5	13	11
r	50	51	52	53	54	55	56	57	58	59
prime	7	13	61	73	13	5	7	13	4561	5
r	60	61	62	63	64	65	66	67	68	69
prime	13	6481	7	13	11	41	13	5	7	13
r	70	71	72	73	74	75	76	77	78	79
prime	271	5	13	41	11	13	31	73	13	5
r	80	81	82	83	84	85	86	87	88	89
prime	7	13	61	5	13	6481	7	13	4561	11
r	90	91	92	93	94	95	96	97	98	99
prime	13	5	7	13	11	5	13	41	7	13
r	100	101	102	103	104	105	106	107	108	109
prime	271	73	13	5	11	13	31	5	13	11
r	110	111	112	113	114	115	116	117	118	119
prime	7	13	61	41	11	5	7	13	4561	5

As a list this is
{13,41,7,13,11,73,13,5,7,13,271,5,13,4561,11,13,31,41,13,5,7,13,61,5,13,41,7,13,4561,11, 13,5,7,13,11,5,13,1181,7,13,271,41,13,5,11,13,31,5,13,117,13,61,73,13,5,7,13,4561,5, 13,6481,7,13,11,41,13,5,7,13,271,5,13,41,11,13,31,73,13,5,7,13,61,5,13,6481,7,13,4561,11, 13,5,7,13,11,5,13,41,7,13,271,73,13,5,11,13,31,5,13,11,7,13,61,41,11,5,7,13,4561,5}
Up to 4*10⁵ only eight values require at least 3000 digits

Value	First Prime	Composite to at least	Filter
95415	5933
128271	4259
201885		9000	3m+2,12m+6
286247	5932
301095		3000	3m,12m+10
343371		3000	3m+1,24m+15
367785		3000	3m+1,12m+6
384815		3000	3m,36m+10

Base 4

Digit 1

s_n^4,1(5) = ((3*5+1)*4^k-1)/3 = (4^k+2-1)/3= (2^k+2-1)(2^k+2-1)/3.

Seed	First Prime	Seed	First Prime	Seed	First Prime	Seed	First Prime
1	1	2	2	3	1	4	1

Note that since s_n^4,1(1) = (2^k+1-1)(2^k+1-1)/3, s_n^4,1(1) is prime only for n=1.

Digit 3

s_n^4,3(3) = ((3+1)*4^k-1 = (4^k+1-1)= (2^k+1-1)(2^k+1-1).

Seed	First Prime	Seed	First Prime
1	1	2	1

Base 5

Digit 1

The prime factors of s_n^5,1(3) have period 2

Residue	1	0
Modulus	2	2
Prime	2	3

The prime divisor list is {3,2}.

Seed	First Prime	Seed	First Prime
1	2	2	1

Digit 2

s_n^5,2(191115) is covered using a period 12 cover

Residue	0	0	3	1	5
Modulus	2	3	4	6	12
Prime	3	31	13	7	601

The primes dividing 12A+r are

0	1	2	3	4	5	6	7	8	9	10	11
3	7	3	31	3	601	3	13	3	31	3	13

The list form of the prime divisors is {3,7,3,31,3,601,3,13,3,31,3,13}.
There are 258 candidates listed at Base5Digit2.html checked to 15000 digits.

Digit 3

s_n^5,3(585655) is covered using a period 12 cover

Residue	1	2	2	0	4
Modulus	2	3	4	6	12
Prime	2	31	13	7	601

The primes dividing 12A+r are

0	1	2	3	4	5	6	7	8	9	10	11
7	2	31	2	601	2	7	2	31	2	13	2

The list of primes is {7,2,31,2,601,2,7,2,31,2,13,2}.
There are 2274 candidates listed at Base5Digit3.html checked to 15000 digits.

Digit 4

s_n^5,4(346801) is covered using a period 12 cover

Residue	1	2	0	0	4
Modulus	2	3	4	6	12
Prime	3	31	13	7	601

The primes dividing 12A+r are

0	1	2	3	4	5	6	7	8	9	10	11
7	3	31	3	13	3	7	3	31	3	601	3

The list of primes is {7,3,31,3,13,3,7,3,31,3,601,3}.
There are 127 candidates listed at Base5Digit4.html checked to 15000 digits.

Base 6

Digit 1

s_n^6,1(26789) is covered using a period 12 cover

Residue	0	0	3	1	5
Modulus	2	3	4	6	12
Prime	7	43	37	31	13

The primes dividing 12A+r are

0	1	2	3	4	5	6	7	8	9	10	11
7	37	7	43	7	13	7	37	7	43	7	37

The list of primes is {7,31,7,43,7,13,7,37,7,43,7,37}.
There are 29 candidate seed values less than 26789 checked to 15000 digits.

value	prime at	composite to	value	prime at	composite to	value	prime at	composite to
50	3008		8443	2700		18592		13000
98	3041		8877	3246		19503	2094
525		13000	9219	4015		19979	3780
575	6476		9288	3826		20518	3150
848	7056		9954		13000	20616		13000
1247		13000	10137		13000	20770		13000
1666	10461		10788		13000	21147	10851
1757	4091		10907		13000	21672	4279
1898		13000	11012		13000	21868		13000
2212	3225		11165	5103		22058	5566
3220	4009		11585		13000	22163	5130
4606	11921		12285	5009		22169	2467
5033	2769		13637	2694		22232		13000
5292	11657		14043	3206		22940		13000
5580	2578		14141	2548		23016		13000
5769	2504		15352		13000	23415	2109
5853	3020		15359	10630		23758		13000
6032	2890		15708	2939		23786	7031
6321	2147		15960		13000	24683		13000
6490	3826		16682		13000	25277		13000
6804	5807		16982	9k-13k		25494		13000
6966	6820		17243		13000	25788		13000
7406		13000	17675	5337		26230	3002
7575	8976		17759		13000	26363	2190
8127		13000	17997	4745		26495		13000

Digit 5

s_n^6,5(84686) is covered using a period 12 cover

Residue	0	1	1	3	11
Modulus	2	4	6	12	12
Prime	7	37	31	13	97

The primes dividing 12A+r are

0	1	2	3	4	5	6	7	8	9	10	11
7	37	7	13	7	37	7	31	7	37	7	97

The list of primes is {7,37,7,13,7,37,7,31,7,37,7,97}.
There are 95 values below 84686 checked to 2000 digits.
1596, 1834, 3589, 4396, 6872, 6986, 8258, 8743, 9366, 9576, 9581, 10694, 11009, 11142, 11389, 14502, 15064, 16828, 17106, 17458, 18284, 19388, 19467, 19906, 20683, 20713, 21539, 21798, 23791, 25051, 26381, 26899, 27447, 28098, 28539, 29846, 30086, 32667, 33626, 35007, 35964, 36771, 37294, 40656, 41237, 41921, 43629, 43993, 44324, 47388, 48384, 48949, 49462, 49553, 49791, 51016, 52463, 53093, 54957, 56201, 57022, 57461, 57491, 57561, 58576, 58756, 59094, 59568, 60648, 61894, 62377, 63028, 63798, 64169, 65002, 66038, 66059, 66302, 66857, 68339, 68857, 70166, 73036, 74186, 76188, 76699, 76706, 77742, 78636, 78958, 79814, 80584, 82516, 83818, 84242. 15 of these are redundant; (1596, 9581, 57491) ,(1834, 11009, 66059), (3589, 21539), (4396, 26381), (6872, 41237), (6986, 41921 ), (8258, 49553), (8743, 52463), (9366, 56201), (9576, 57461), (10694, 64169), (11142, 66857 ), (11389, 68339)
leaving 80 candidates for smallest seed.
1596, 1834, 3589, 4396, 6872, 6986, 8258, 8743, 9366, 9576, 10694, 11142, 11389, 14502, 15064, 16828, 17106, 17458, 18284, 19388, 19467, 19906, 20683, 20713, 21798, 23791, 25051, 26899, 27447, 28098, 28539, 29846, 30086, 32667, 33626, 35007, 35964, 36771, 37294, 40656, 43629, 43993, 44324, 47388, 48384, 48949, 49462, 49791, 51016, 53093, 54957, 57022, 57561, 58576, 58756, 59094, 59568, 60648, 61894, 62377, 63028, 63798, 65002, 66038, 66302, 68857, 70166, 73036, 74186, 76188, 76699, 76706, 77742, 78636, 78958, 79814, 80584, 82516, 83818, 84242.
There are 34 candidate seed values less that 84686 remaining after checking to 12000 digits.
Most of these have been eliminated. Some of the prime are available at Gary Barnes's Riesel Conjectures page.

value	prime at	composite to	value	prime at	composite to	value	prime at	composite to
1596		1210000	26899	4542		58756	CRUS
1834	5533		27447	2135		59094	171929
3589	9186		28098	2355		59568	5996
4396	4175		28539	2545		60648	3857
6872	2268		29846	141526		61894	6215
6986	5119		30086	5437		62377	2559
8258	CRUS		32667	2122		63028	7317
8743	5417		33626	CRUS		63798	5821
9366	7845		35007	CRUS		65002	8507
9576	121099		35964	CRUS		66038	CRUS
10694	CRUS		36771		1210000	66302	3204
11142	3744		37294	CRUS		68857	7982
11389	CRUS		40656	CRUS		70166	4272
14502	4680		43629	5720		73036	7676
15064	2117		43993	569498		74186	CRUS
16828	CRUS		44324	2451		76188	7865
17106	6528		47388	4246		76699	8395
17458	CRUS		48384	6653		76706	CRUS
18284	CRUS		48949	143236		77742	560745
19388	CRUS		49462	CRUS		78636	4160
19467	3529		49791	CRUS		78958	458114
19906	CRUS		51016	528803		79814	113777
20683	CRUS		53093	2844		80584	2201
20713	3979		54957	CRUS		82516	2517
21798	CRUS		57022	483561		83818	3197
23791	2054		57561	2439		84242	3540
25051	2842		58576	7185

Conjectures 'R Us has checked most of these. The prime values for greater than 12000 digits are from CRUS.

Base 7

Digit 1

s_n^7,1(76) uses a period 12 cover to show that all terms are composite.

Residue	0	2	0	3	1
Period	2	3	3	4	12
Prime	2	3	19	5	13

Primes dividing s_12A+r^7,1(76) are

r	0	1	2	3	4	5	6	7	8	9	10	11
prime	2	13	2	19	2	3	2	5	2	19	2	3

The list of primes is {2,13,2,19,2,3,2,5,2,19,2,3}.
The smaller seed values produce primes

Seed	First Prime	Seed	First Prime	Seed	First Prime	Seed	First Prime	Seed	First Prime
1		2	3	3	4	4	1	5	18
6	1	7	4	8	3	9	2	10	1
11	2	12	127	13	424	14	5	15	1
16	1	17	20	18	3	19	256	20	4
21	4	22	3	23	468	24	5	25	10
26	9	27	14	28	1	29	2	30	1
31	4	32	27	33	4	34	1	35	2
36	17	37	4	38	5	39	40	40	1
41	2	42	31	43	4	44	3	45	2
46	45	47	2	48	1	49	6	50	3
51	260	52	5907	53	8	54	1	55	46
56	177	57	2	58	177	59	18	60	1
61	15118	62	3	63	16	64	1	65	938
66	1	67	4	68	15	69	2	70	1
71	8	72	7	73	4	74	3	75	398

Digit 2

s_n^7,2(15979) uses a period 12 cover to show that all terms are composite.

Residue	1	0	1	2	11
Period	3	3	4	6	12
Prime	3	19	5	43	13

Primes dividing s_30A+r^7,2(15979) are

r	0	1	2	3	4	5	6	7	8	9	10	11
prime	19	3	43	19	3	5	19	3	43	19	3	13

The list of primes is {19,3,43,19,3,5,19,3,43,19,3,13}.

31 candidates through 3000 digits.
(2047, 14331), (2261, 15829), 3601, 3667, 4011, 4101, 4315, 4785, 5149, 5507, 5849, 6479, 6609, 7505, 8305, 8911, 8937, 9069, 9331, 9839, 10279, 11837, 12347, 12565, 12979, 14331, 14759, 14945, 15181, 15711,.
Removing redundancies leaves 29 candidates less than 28521 that are checked to 3000 digits. 2047, 2261, 3601, 3667, 4011, 4101, 4315, 4785, 5149, 5507, 5849, 6479, 6609, 7505, 8305, 8911, 8937, 9069, 9331, 9839, 10279, 11837, 12347, 12565, 12979, 14759, 14945, 15181, 15711.
There are 17 candidate seed values less that 28521 remaining after checking to 15000 digits.
13096

value	prime at	composite to	value	prime at	composite to	value	prime at	composite to
2047		15000	5849	6084		10279	3074
2261			6479		15000	11837	14782
3601		15000	6609		15000	12347		15000
3667		15000	7505		15000	12565	3159
4011		15000	8305		15000	12979	4836
4101		15000	8911	3839		14759		15000
4315	5502		8937	3242		14945		15000
4785		15000	9069	9263		15181		15000
5149		15000	9331		15000	15711	7429
5507	12597		9839		15000

Digit 3

s_n^7,3(5629) uses a period 12 cover to show that all terms are composite.

Residue	0	1	2	2	0
Period	2	3	4	6	12
Prime	2	19	5	43	13

Primes dividing s_12A+r^7,3(5629) are

r	0	1	2	3	4	5	6	7	8	9	10	11
prime	13	2	5	2	19	2	5	2	43	2	5	2

The list of primes is {13,2,5,2,19,2,5,2,43,2,5,2}.

54 candidates less than 5629 checked through 2000 digits.
(98,689,4826) (214, 1501) (358,2509), (515,3608), (556,3895), (589,4126) (734,5141), 896, 905, 989, 1126, 1246, 1336, 1516, 1639, 1796, 1945,2168, 2323, 2465, 2468, 2695, 2848, 2885, 3028, 3056, 3245, 3416, 3518, 3628, 3875, 4145, 4171, 4256, 4258, 4565, 4579, 4586, 4736, 4828, 4856, 5095, 5116, 5366, 5510, 5546, 5629
There are 46 candidates for smallest seed below 5629 checked through 2000 digits.
98, 214, 358, 515, 556, 589, 734, 896, 905, 989, 1126, 1246, 1336, 1516, 1639, 1796, 1945, 2168, 2323, 2465, 2468, 2695, 2848, 2885, 3028, 3056, 3245, 3416, 3518, 3628, 3875, 4145, 4171, 4256, 4258, 4565, 4579, 4586, 4736, 4828, 4856, 5095, 5116, 5366, 5510, 5546.
raising the check to 15000 digits leaves 19 smaller seed candidates.

value	prime at	composite to	value	prime at	composite to	value	prime at	composite to
98		15000	1945		15000	4145	14802
214	3815		2168		15000	4171	8992
358		15000	2323	2964		4256		15000
515		15000	2465		15000	4258	3885
556	4895		2468		15000	4565		15000
589		15000	2695	4882		4579		15000
734	4669		2848	5589		4586		15000
896	5839		2885		15000	4736	2931
905		15000	3028		15000	4828		15000
989		15000	3056	2139		4856	3503
1126		15000	3245		15000	5095		15000
1246	2567		3416	3955		5116		15000
1336		15000	3518		15000	5366	4583
1516	2819		3628		15000	5510	5911
1639	2024		3875	2894		5546		15000
1796	3307

Digit 4

s_n^7,4(20277) uses a period 12 cover to show that all terms are composite.

Residue	0	2	2	1	5
Period	3	3	4	6	12
Prime	3	19	5	43	13

Primes dividing s_12A+r^7,4(20277) are

r	0	1	2	3	4	5	6	7	8	9	10	11
prime	3	43	19	3	13	19	3	43	19	3	5	19

The list of primes is {3,43,19,3,31,19,3,43,19,3,5,19}.

39 candidates checked through 2000 digits.
(507,3553),(2217,15523),(2303,16125), 3059,3405,3775,4203,4469,4515,4915,6517,6739,8687,10397, 10697,10813,12087,12293,12635,13095,13361,14583,14615,14801,14953,15599, 16211,16511,17313,18107,18139,18145,18863,19675,19703,20229,20277
There are 36 candidates less than 20277 checked through 2000 digits.
507,2217,2303,3059,3405,3775,4203,4469,4515,4915, 6517,6739,8687,10397,10697,10813,12087,12293,12635,13095, 13361,14583,14615,14801,14953,15599,16211,16511,17313,18107, 18139,18145,18863,19675,19703,20229.
Checking through 15000 digits leave 16 smaller seed candidates.

value	prime at	composite to	value	prime at	composite to	value	prime at	composite to
507		15000	8687		15000	14953		15000
2217	3344		10397		15000	15599	4526
2303	2535		10697		15000	16211	3891
3059	3077		10813	13434		16511		15000
3405		15000	12087	6805		17313	4210
3775	4449		12293		15000	18107		15000
4203		15000	12635		15000	18139	10842
4469		15000	13095	2011		18145	9631
4515	2245		13361	2841		18863	2247
4915	7446		14583	2098		19675		15000
6517		15000	14615	2949		19703		15000
6739		15000	14801	3461		20229	2426

Digit 5

s_n^7,5(43) uses a period 6 cover to show that all terms are composite.

Residue	0	0	1	5
Period	2	3	3	6
Prime	2	3	19	43

Primes dividing s_12A+r^7,5(43) are

r	0	1	2	3	4	5
prime	2	19	2	3	2	43

The list of primes is {2,19,2,3,2,43}.

Seed	First Prime	Seed	First Prime	Seed	First Prime	Seed	First Prime
1	2	2	1	3	4	4	3
6	1	7	2	8	1	9	4
11	4	12	1	13	2	14	1
16	5	17	10	18	1	19	2
21	2	22	5	23	4	24	1
26	3	27	14	28	9	29	6
31	2	32	1	33	2	34	5
36	1	37	6	38	1	39	2
41	10	42	7	43	never	44	1

Digit 6

2384410917857141 uses a period of cover 60 s_n^7,6(2384410917857141) uses a period 60 cover to show that all terms are composite.

Residue	2	3	0	4	2	6	9	1	9	18	48
Period	3	4	5	6	10	10	12	12	15	20	60
Prime	19	5	2801	43	11>/TD>	191	13	181	31	281	61

The primes dividing 60A + r are

r	0	1	2	3	4	5	6	7	8	9
prime	2801	181	19	5	43	19	191	5	19	13
r	10	11	12	13	14	15	16	17	18	19
prime	2801	19	11	181	19	5	43	19	281	5
r	20	21	22	23	24	25	26	27	28	29
prime	19	13	11	19	31	2801	19	5	43	19
r	30	31	32	33	34	35	36	37	38	39
prime	2801	5	19	13	43	19	191	181	19	5
r	40	41	42	43	44	45	46	47	48	49
prime	2801	19	11	5	19	2801	43	19	61	181
r	50	51	52	53	54	55	56	57	58	59
prime	19	5	43	19	31	5	19	13	43	19

There are 4 values up to 10⁵ checked through 3000 digits.
48251, 48583, 78647, 92017 Need to check 12m+(0;2;5;6;9), 12m+(5;8;11), 12m+(3;4;10),12m+(4; 7;8; 10;11), resp.

Base 8

Digit 1

s_n^8,1(21) uses a period 12 cover to show that all terms are composite.

Residue	0	2	2
Period	2	4	4
Prime	3	5	13

Primes dividing s_12A+r^8,1(21) are

r	0	1	2	3
prime	3	13	3	5

The list of primes is {3,13,3,5}.

Seed	First Prime	Seed	First Prime	Seed	First Prime	Seed	First Prime
1	2	2	1	3	3	4	12
5	1	6	21	7	2	8	2
9	1	10	6	11	1	12	1
13	314	14	1	15	3	16	2
17	1	18	139	19	20	20	2

Digit 3

s_n^8,3(1079770) uses a period 120 cover to show that all terms are composite.

Residue	1	0	3	0	5	1	7	2
Period	3	4	4	6	8	8	10	12
Prime	73	5	13	19	17	241	11	37

Primes dividing s_120A+r^8,3(1079770) are

r	0	1	2	3	4	5	6	7	8	9
prime	13	41	7	13	11	73	13	5	7	13
r	10	11	12	13	14	15	16	17	18	19
prime	271	5	13	4561	11	13	31	41	13	5
r	20	21	22	23	24	25	26	27	28	29
prime	7	13	61	5	13	41	7	13	4561	11
r	30	31	32	33	34	35	36	37	38	39
prime	13	5	7	13	11	5	13	1181	7	13
r	40	41	42	43	44	45	46	47	48	49
prime	271	41	13	5	11	13	31	5	13	11
r	50	51	52	53	54	55	56	57	58	59
prime	7	13	61	73	13	5	7	13	4561	5
r	60	61	62	63	64	65	66	67	68	69
prime	13	6481	7	13	11	41	13	5	7	13
r	70	71	72	73	74	75	76	77	78	79
prime	271	5	13	41	11	13	31	73	13	5
r	80	81	82	83	84	85	86	87	88	89
prime	7	13	61	5	13	6481	7	13	4561	11
r	90	91	92	93	94	95	96	97	98	99
prime	13	5	7	13	11	5	13	41	7	13
r	100	101	102	103	104	105	106	107	108	109
prime	271	73	13	5	11	13	31	5	13	11
r	110	111	112	113	114	115	116	117	118	119
prime	7	13	61	41	11	5	7	13	4561	5

The list of primes is {3,43,19,3,31,19,3,43,19,3,5,19}.

There are 721 candidates less than 1079770 listed at Base8Digit3.html checked to 15000 digits.

Digit 5

s_n^8,5(7476) uses a period 8 cover to show that all terms are composite.

Residue	0	1	3	7
Period	2	4	8	8
Prime	3	13	17	241

Primes dividing s_8A+r^8,5(7476) are

r	0	1	2	3	4	5	6	7
prime	3	13	3	17	3	13	3	241

The list of primes is {3,13,3,17,3,13,3,241}.

There are 29 candidates for minimum seed less than 7476 checked to 15000 digits.

value	prime at	composite to	value	prime at	composite to	value	prime at	composite to
1241		15000	3458		15000	5703	7675
1673		15000	3503		15000	6267		15000
1791		15000	3762	8529		6312		15000
1898		15000	3801		15000	6611		15000
2046		15000	3843	7659		6851		15000
2171		15000	(486,3893)		15000	6896	13840
2274		15000	4121		15000	7014		15000
2336		15000	4283		15000	7232		15000
2489		15000	(539,4317)		15000	7233		15000
2553	8449		4464		15000	7272	7665
2723	11488		5058	7295		7398		15000
2982		15000	5292		15000
2999	7828		5531		15000
3116	10796		5627		15000

Digit 7

s_n^8,7(13) uses a period 4 cover to show that all terms are composite.

Residue	0	1	3
Period	2	4	4
Prime	3	5	13

Primes dividing s_4A+r^8,7(13) are

r	0	1	2	3
prime	3	5	3	13

The list of primes is {3,5,3,13}.

Seed	First Prime	Seed	First Prime	Seed	First Prime	Seed	First Prime
1	2	2	1	3	1	4	4
5	1	6	3			8	1
9	1	10	18	11	3	12	1

Base 9

Digit 1

111...1 = (9^k-1)/8 = (3^k-1)(3^k+1)/(2*4)

Digit 2

s_n^9,2(4615) uses a period 6 cover to show that all terms are composite.

Residue	0	2	0	1
Period	2	3	3	6
Prime	5	7	13	73

Primes dividing s_8A+r^9,2(4615) are

r	0	1	2	3	4	5
prime	5	73	5	13	5	7

The list of primes is {5,73,5,13,5,7}.

The 14 candidates less than 4615 have been checked to 15000 digits.

value	prime at	composite to	value	prime at	composite to
307		15000
847		15000	1945		15000
975		15000	2555	6185
1125		15000	2995		15000
1157		15000	3157		15000
1197	7024		3437	9434
1255		15000	3985		15000
1637	11336		4157	5262
1679		15000	4167		15000
1787		15000	4195		15000

Digit 4

s_n^9,4(6059) uses a period 6 cover to show that all terms are composite.

Residue	1	2	1	0
Period	2	3	3	6
Prime	5	7	13	73

Primes dividing s_8A+r^9,4(6059) are

r	0	1	2	3	4	5
prime	73	5	7	5	13	5

The list of primes is {73,5,7,5,13,5}.

The 12 remaining candidate seeds less than 6059 checked to 18000 digits.

value	prime at	composite to	value	prime at	composite to
915		18000	3755	13011
1495	10423		3813		18000
1565		18000	3969		18000
1635	2479		3975	2041
2419		18000	4029		18000
2877		18000	4225	12379
2905		18000	4905		18000
3059	14290		5325		18000
3393		18000	5383		18000
3689	8428		5535	2045
			5625	12157

Most of these are covered in the Base 3 digit 1 case. So the searches can be combined.

Digit 5

s_n^9,5(78) uses a period 6 cover to show that all terms are composite.

Residue	0	1	0	5
Period	2	3	3	6
Prime	2	7	13	73

Primes dividing s_8A+r^9,5(78) are

r	0	1	2	3	4	5
prime	2	7	2	13	2	73

The list of primes is {2,7,2,13,2,73}.

Seed	First Prime	Seed	First Prime	Seed	First Prime	Seed	First Prime
1	2	2	1	3	2	4	1
6	1	7	2	8	3	9	4
11	2	12	1	13	2	14	1
16	1	17	2	18	1	19	442
21	14	22	3	23	2	24	5
26	1	27	2	28	1	29	2
31	4	32	1	33	4	34	1
36	3	37	264	38	1	39	2
41	2	42	1	43	2	44	1
46	1	47	10	48	3	49	2
51	12	52	5	53	6	54	1
56	1	57	282	58	3	59	4
61	4	62	1	63	2	64	3
66	1	67	2	68	1	69	2
71	2	72	1	73	10	74	3
76	21	77	2

Digit 7

s_n^9,7(2) uses a period 2 cover to show that all terms are composite.

Residue	1	0
Period	2	2
Prime	2	5

Primes dividing s_2A+r^9,7(2) are

r	0	1
prime	5	2

The list of primes is {5,2}.

177₉= 151₁₀ is prime.

Digit 8

s_n^9,8(3)= (4*9^k-1)= (2*3^k-1)*(2*3^k+1).
18₉ = 17₁₀ is prime.

Base 10

Digit 1

Lenny Jones proved that 37 is the minimal seed and produced the table showing the first appearance of a prime for each smaller seed.
s_n^10,1(37) uses a period 6 cover to show that all terms are composite.

Residue	0	1	0	5
Period	2	3	3	6
Prime	2	7	13	73

Primes dividing s_8A+r^10,1(37) are

r	0	1	2	3	4	5
prime	7	3	37	13	3	37

The list of primes is {7,3,37,13,3,37}.

Digit 3

The only candidate seed less that 4070 is 817. This has been checked to 554789 digits.
Candidates requiring at least 2500 digits to produce a prime are

seed	digits
410	37398
817
1037	3292
1166	12689
1279	4752
2959	6763
3674	16097

Digit 7

The minimal seed is 891.

Digit 9

Two candidates less than 10175 remain.

value	prime at	composite to
449	11958
1343	29711
1802	45882
1934	51836
3355	13323
4015	3647
4420		630000
4477	4817
6587	5846
6664	60248
7018		630000
8578	373260

H. Riesel, Några stora primtal (Swedish: Some large primes), Elementa 39 (1956), 258-260.
L. Jones, When does appending the same digit repeatedly on the right of a positive integer generate a sequence of composite numbers?, Amer. Math. Monthly 118 (2011), 153-160.
The Reisel problem
The repunit case for other bases OEIS A084740
Karsten Bonath's Riesel Prime database No Prime left behind summary A condensed table for the extended Riesel conjectures at No Prime Left Behind