Prime forms involving Repunits.

Sun Oct 17 03:23:33 CEST 2004

Et al,

	Here is where we are today. All of the sequences below have been tested to an 
exponent of five thousand. I have changed all titles to the use of R_n in place
of (10^n-1)/9. The second definition, which is shorter, I put it into the first
%Comment line. This is the first half of the sequences that I mailed out Thursday
14 October 2004.

	I found that the web site of Prof. Kamada at
http://homepage2.nifty.com/m_kamada/math/ to be most helpful.

the index should have a category for:

Prime forms involving Repunits, R_n.
	 X*Repunits+/-Y, (X,Y)=1, X+/-Y <10, X&Y belong to {d} d being the digits 1..9.

A004023, A097683, A097684, A097685, A084832, A096506, A099409, A099410, A055557, 
A099411,
A099412, A096845, A099413, A099414, A099415, A099416, A099417, A099418, A098088, 
A096507,
A099419, A099420, A098089, A099421, A099422, A096846, A096508, A095714, A089675

      X*10*Repunits+Y, (X,Y)=1, X&Y belong to {d} d being the digits 1..9.

A004023, A056654, A056655, A056659, A056660, A056656, A056677, A056678, A055520, 
A056680,
A056681, A056661, A056682, A056683, A056684, A056685, A056686, A056687, A056658, 
A056657,
A056688, A056689, A056693, A056664, A056694, A056695, A056663, A056696, A056662.

	Except for the initial sequence, A004023 Prime "repunits", all others match
up such that the terms in the first group less one is equal to the terms in the
second group.

	I would like to see in the %Hyper-link section the following:
%H  <a href="http://www.research.att.com/~njas/sequences/Sindx_Pri.html">Index 
entries for Primes forms involving Repunits</a>
instead of listing one of the two lists above.

Form 11..11
%I A004023 M2114
%S A004023 2,19,23,317,1031,49081,86453
%N A004023 Prime "repunits": 11...111 = R_n = (10^n - 1)/9 is prime.
............
%H A004023 M. Kamada, <a 
href="http://homepage2.nifty.com/m_kamada/math/11111.htm">Factorizations of 
11...11 (Repunit)</a>.
............
%Y A004023 See A004022 for the actual primes.
%K A004023 hard,nonn,nice
%O A004023 1,1
%A A004023 njas

Form 111..113
%I A097683
%S A097683 0,1,2,3,5,9,11,24,84,221,1314,,2952
%N A097683 Numbers n such that R_n + 2 is prime, where R_n = 11...1 is the repunit 
(A002275) of length n.
%C A097683 Also numbers n such that (10^n+17)/9 is prime.
%C A097683 Values indicate primes of the form "(n-1) ones followed by a three"; 
zero is a degenerate case. Related to the base 10 repunit primes.
%t A097683 Do[ If[ PrimeQ[(10^n - 1)/9 + 2], Print[n]], {n, 0, 5951}] (from RGWv 
Oct 15 2004)
%Y A097683 Cf. A004023, A097684, A097685.
%K A097683 hard,more,nonn
%O A097683 0,3
%A A097683 Carl R. White (cyrek(AT)cyreksoft.yorks.com), Aug 19 2004
%E A097683 a(10) from RGWv (rgwv at rgwv.com), Oct 15 2004

%I A056654
%S A056654 0,1,2,4,8,10,23,83,220,1313,2951
%N A056654 Numbers n such that 10*R_n + 3 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A056654 Also numbers n such that (10^(n+1)+17)/9 is prime.
%t A056654 Do[ If[ PrimeQ[ 10*(10^n - 1)/9 + 3 ], Print[n]], {n, 5951}]
%Y A056654 Cf. A093011 (corresponding primes).
%K A056654 hard,more,nonn
%O A056654 0,3
%A A056654 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 09 2000
%E A056654 a(10) (only a probable prime) from Rick L. Shepherd 
(rshepherd2(AT)hotmail.com), Mar 14 2004

Form 111..117
%I A097684
%S A097684 1,2,4,5,8,23,29,40,131,136,215,611,767,2153,2576
%N A097684 Numbers n such that R_n + 6 is prime, where R_n = 11...1 is the repunit 
(A002275) of length n.
%C A097684 Also numbers n such that (10^n+53)/9 is prime.
%C A097684 Values indicate primes of the form "(n-1) ones followed by a seven". 
Related to the base 10 repunit primes.
%C A097684 2153 and 2576 produce probable primes. - a(12)-a(15) from Rick L. 
Shepherd (rshepherd2(AT)hotmail.com), Aug 23 2004
%F A097684 a(n) = A056655(n) + 1 for all n >= 0. - a(12)-a(15) from Rick L. 
Shepherd (rshepherd2(AT)hotmail.com), Aug 23 2004
%o A097684 (PARI) for (i=1,1000,if(isprime((10^i-1)/9 + 
6),print1(i,","),print1("."))) (Bouayoun)
%t A097684 Do[ If[ PrimeQ[(10^n - 1)/9 + 6], Print[n]], {n, 0, 5000}] (from RGWv 
Oct 14 2004)
%Y A097684 Cf. A004023, A056655, A097683, A097685.
%K A097684 hard,more,nonn
%O A097684 0,2
%A A097684 Carl R. White (cyrek(AT)cyreksoft.yorks.com), Aug 19 2004
%E A097684 a(12) and a(13) from Mohammed Bouayoun 
(mohammed.bouayoun(AT)sanef.com), Aug 23 2004.
%E A097684 a(12)-a(15) from Rick L. Shepherd (rshepherd2(AT)hotmail.com), Aug 23 2004

%I A056655
%S A056655 0,1,3,4,7,22,28,39,130,135,214,610,766,2152,2575
%N A056655 Numbers n such that 10*R_n + 7 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A056655 Also numbers n such that (10^(n+1)+53)/9 is prime.
%C A056655 2575 also produces a probable prime.
%F A056655 a(n) = A097684(n) - 1 for all n >= 0. - Rick L. Shepherd 
(rshepherd2(AT)hotmail.com), Aug 23 2004
%t A056655 Do[ If[ PrimeQ[ 10*(10^n - 1)/9 + 7 ], Print[ n ]], {n, 5000}]
%Y A056655 Cf. A093139 (corresponding primes).
%Y A056655 Cf. A097684.
%K A056655 hard,nonn
%O A056655 0,3
%A A056655 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 09 2000
%E A056655 2152 (giving a probable prime) from Rick L. Shepherd 
(rshepherd2(AT)hotmail.com), Mar 23 2004
%E A056655 2575 from Rick L. Shepherd (rshepherd2(AT)hotmail.com), Aug 23 2004

Form 111..119
%I A097685
%S A097685 2,5,6,8,17,50,684,720,1452,1679,3146
%N A097685 Numbers n such that R_n + 8 is prime, where R_n = 11...1 is the repunit 
(A002275) of length n.
%C A097685 Also numbers n such that (10^n+71)/9 is prime.
%C A097685 Equals A056659 + 1. See A056659, the main entry for this problem, for 
additional information.
%C A097685 Values indicate primes of the form "(n-1) ones followed by a nine". 
Related to the base 10 repunit primes.
%t A097685 Do[ If[ PrimeQ[(10^n - 1)/9 + 8], Print[n]], {n, 0, 5000}] (from RGWv 
Oct 14 2004)
%Y A097685 Equals A056659 + 1. Cf. A004023, A097683, A097684.
%K A097685 nonn
%O A097685 1,1
%A A097685 Carl R. White (cyrek(AT)cyreksoft.yorks.com), Aug 19 2004

%I A056659
%S A056659 1,4,5,7,16,49,683,719,1451,1678,3145
%N A056659 Numbers n such that 10*R_n + 9 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A056659 Also numbers n such that (10^(n+1)+71)/9 is prime.
%t A056659 Do[ If[ PrimeQ[10*(10^n - 1)/9 + 9], Print[n]], {n, 5000}]
%Y A056659 See A097685 for another version.
%K A056659 hard,nonn
%O A056659 0,2
%A A056659 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 09 2000

Form 222..221
%I A084832
%S A084832 4,18,100,121,244,546,631,1494,2566
%N A084832 Numbers n such that 2*R_n - 1 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A084832 Also numbers n such that (2*10^n-11)/9 is prime.
%C A084832 Larger values not certified prime.
%e A084832 a(1)=4 because 2*(10^4-1)/9-1 = 2221 is prime.
%e A084832 n=18 means that 222222222222222221 is prime.
%t A084832 Do[ If[ PrimeQ[(10^n - 1)/9 + 6], Print[n]], {n, 0, 7000}] (from RGWv 
Oct 14 2004)
%Y A084832 Cf. A084831, A096503, A096504, A096505, A096506, A096507, A096508, 
A096841, A096842, A096843, A096844, A096845, A096846, A000203.
%K A084832 more,nonn
%O A084832 1,1
%A A084832 Jason Earls (jcearls(AT)cableone.net), Jun 05 2003
%E A084832 a(8) from Labos E. (labos(AT)ana1.sote.hu), Jul 15 2004

%I A096844 is the same as A084832, so simply delete A096844.

%I A056660
%S A056660 3,17,99,120,243,545,630,1493,2565
%N A056660 Numbers n such that 20*R_n + 1 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A056660 Also numbers n such that (2*10^(n+1)-11)/9 is prime.
%t A056660 Do[ If[ PrimeQ[ 20*(10^n - 1)/9 + 1 ], Print[n]], {n, 7000}]
%Y A056660 Cf. A091189 (corresponding primes).
%K A056660 nonn
%O A056660 0,1
%A A056660 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 09 2000
%E A056660 a(8) and a(9) from Rick L. Shepherd (rshepherd2(AT)hotmail.com), Feb 22 
2004

Form 222..223
%I A096506
%S A096506 1,2,3,8,11,36,95,101,128,260,351,467,645,1011,1178,1217,2442,3761,3806
%N A096506 Numbers n for which 2*R_n + 1 is a prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A096506 Also numbers n such that (2*10^n+7)/9 is prime.
%e A096506 n=36: 222222222222222222222222222222222223 is a prime number.
%t A096506 Do[ If[ PrimeQ[ 2(10^n - 1)/9 + 1], Print[n]], {n, 7000}] (from RGWv 
Oct 14 2004)
%Y A096506 Cf. A096503, A096504, A096505, A096506, A096507, A096508.
%K A096506 base,nonn
%O A096506 1,2
%A A096506 Labos E. (labos(AT)ana1.sote.hu), Jul 12 2004

%I A056656
%S A056656 0,1,2,7,10,35,94,100,127,259,350,466,644,1010,1177,1216,2441,3760,3805
%N A056656 Numbers n such that 20*R_n + 3 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A056656 Also numbers n such that (2*10^(n+1)+7)/9 is prime.
%C A056656 2441 produces a probable prime. - 2442 from Rick L. Shepherd 
(rshepherd2(AT)hotmail.com), Mar 27 2004
%t A056656 Do[ If[ PrimeQ[ 20*(10^n - 1)/9 + 3 ], Print[n]], {n, 7000}]
%Y A056656 Cf. A093162 (corresponding primes).
%K A056656 hard,nonn
%O A056656 0,3
%A A056656 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 09 2000
%E A056656 2442 from Rick L. Shepherd (rshepherd2(AT)hotmail.com), Mar 27 2004

Form 222..227
%I A099409
%S A099409 1,3,9,15,28,64,1168,1695,2362
%N A099409 Numbers n such that 2*R_n + 5 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A099409 Also numbers n such that (2*10^n+43)/9 is prime.
%t A099409 Do[ If[ PrimeQ[ 2(10^n - 1)/9 + 5], Print[n]], {n, 5000}]
%Y A099409 Cf. .
%O A099409 0,2
%K A099409 nonn
%A A099409 Robert G. Wilson v (rgwv at rgwv.com), Oct 14 2004

%I A056677
%S A056677 0,2,8,14,27,63,1167,1694,2361
%N A056677 Numbers n such that 20*R_n + 7 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A056677 Also numbers n such that (2*10^(n+1)+43)/9 is prime.
%t A056677 Do[ If[ PrimeQ[20*(10^n - 1)/9 + 7], Print[n]], {n, 0, 5000}]
%K A056677 hard,nonn
%O A056677 0,2
%A A056677 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 10 2000

Form 222..229
%I A099410
%S A099410 2,3,5,14,176,416,2505,2759
%N A099410 Numbers n such that 2*R_n + 7 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A099410 Also numbers n such that (2*10^n+61)/9 is prime.
%t A099410 Do[ If[ PrimeQ[ 2(10^n - 1)/9 + 7], Print[n]], {n, 5000}]
%Y A099410 Cf. .
%O A099410 0,2
%K A099410 nonn
%A A099410 Robert G. Wilson v (rgwv at rgwv.com), Oct 14 2004

%I A056678
%S A056678 1,2,4,13,175,415,2504,2758
%N A056678 Numbers n such that 20*R_n + 9 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A056678 Also numbers n such that (2*10^(n+1)+61)/9 is prime.
%t A056678 Do[ If[ PrimeQ[20*(10^n - 1)/9 + 9], Print[n]], {n, 5000}]
%Y A056678 Cf. A093401 (corresponding primes).
%K A056678 hard,nonn
%O A056678 1,2
%A A056678 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 10 2000
%E A056678 Two more terms from Rick L. Shepherd (rshepherd2(AT)hotmail.com), Mar 
30 2004

Form 333..331
%I A055557
%S A055557 2,3,4,5,6,7,8,18,40,50,60,78,101,151,319,382,784,1732,1918,8855,11245,
%T A055557 11960,12130,18533,22718,23365,24253,24549,25324,30178
%N A055557 Numbers n such that 3*R_n - 2 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A055557 Also numbers n such that (10^n-7)/3 is prime.
%C A055557 Sierpinski attributes the primes for n = 2,...,8 to A. Makowski.
%D A055557 W. Sierpinski, 200 Zadan z Elementarnej Teorii Liczb [200 Problems from 
the Elementary Theory of Numbers], Warszawa, 1964; Problem 88.
%H A055557 Dave Rusin, <a 
href="http://www.math.niu.edu/~rusin/known-math/98/exp_primes">Primes in 
exponential sequences</a>
%t A055557 Do[ If[ PrimeQ[(10^n - 7)/3], Print[n]], {n, 50410}]
%o A055557 (PARI) for(n=1,2000, if(isprime((10^n-7)/3),print(n)))
%Y A055557 Cf. A051200, A033175.
%K A055557 nonn
%O A055557 2,1
%A A055557 Labos E. (labos(AT)ana1.sote.hu), Jul 10 2000
%E A055557 Corrected and extended by Jason Earls (jcearls(AT)cableone.net), Sep 22 
2001

%I A055520
%S A055520 1,2,3,4,5,6,7,17,39,49,59,77,100,150,318,381,783,1731,1917,8854,11244,
%T A055520 11959,12129,18532,22717,23364,24252,24548,25323,30177
%N A055520 Numbers n such that 30*R_n + 1 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A055520 Also numbers n such that (10^(n+1)-7)/3 is prime.
%D A055520 Martin Gardner, Strong Laws of Small Primes, in The Last Recreations, 
p. 194 (1997).
%F A055520 A055557 plus 1.
%H A055520 E. W. Weisstein, <a href="http://mathworld.wolfram.com/3.html">Link to 
a section of The World of Mathematics.</a>
%t A055520 Do[ If[ PrimeQ[30*(10^n - 1)/9 + 1], Print[n]], {n, 0, 50410}]
%Y A055520 Indices of A033175 that are prime. Cf. A051200, A055557.
%K A055520 hard,nonn
%O A055520 1,2
%A A055520 Eric W. Weisstein (eric(AT)weisstein.com)
%E A055520 No others with n <= 4000.

Form 333..337
%I A099411
%S A099411 1,2,3,6,46,394,978,2586,2811,2968,3642,4827,4918,5592,5706
%N A099411 Numbers n such that 3*R_n + 4 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A099411 Also numbers n such that (10^n+11)/3 is prime.
%t A099411 Do[ If[ PrimeQ[ 3(10^n - 1)/9 + 4], Print[n]], {n, 10000}]
%Y A099411 Cf. .
%O A099411 1,2
%K A099411 hard,nonn
%A A099411 Robert G. Wilson v (rgwv at rgwv.com), Oct 14 2004

%I A056680
%S A056680 0,1,2,5,45,393,977,2585,2810,2967,3641,4826,4917,5591,5705
%N A056680 Numbers n such that 30*R_n + 7 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A056680 Also numbers n such that (10^(n+1)+11)/3 is prime.
%t A056680 Do[ If[ PrimeQ[30*(10^n - 1)/9 + 7], Print[n]], {n, 0, 10000}]
%K A056680 hard,nonn
%O A056680 0,3
%A A056680 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 10 2000

Form 444..441
%I A099412
%S A099412 0,2,4,11,28,55,94,475,2080,4835,5845
%N A099412 Numbers n such that 4*R_n - 3 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A099412 Also numbers n such that (4*10^n-31)/9 is prime.
%t A099412 Do[ If[ PrimeQ[ 4(10^n - 1)/9 - 3], Print[n]], {n, 0, 7000}]
%Y A099412 Cf. .
%O A099412 0,2
%K A099412 nonn
%A A099412 Robert G. Wilson v (rgwv at rgwv.com), Oct 14 2004

%I A056681
%S A056681 1,3,10,27,54,93,474,2079,4834,5844
%N A056681 Numbers n such that 40*R_n + 1 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A056681 Also numbers n such that (4*10^(n+1)-31)/9 is prime.
%t A056681 Do[ If[ PrimeQ[40*(10^n - 1)/9 + 1], Print[n]], {n, 0, 7000}]
%K A056681 hard,nonn
%O A056681 0,2
%A A056681 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 10 2000

Form 444..443
%I A096845
%S A096845 1,2,3,6,9,12,30,32,183,297,492
%N A096845 Numbers n for which 4*R_n - 1 is a prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A096845 Also numbers n such that (4*10^n-13)/9 is prime.
%e A096845 n=30 means that 444444444444444444444444444443 is prime.
%t A096845 Do[ If[ PrimeQ[ 4(10^n - 1)/9 - 1], Print[n]], {n, 5000}] (from RGWv 
Oct 14 2004)
%Y A096845 Cf. A096503-A096508, A096841-A096846, A000203.
%K A096845 hard,more,nonn
%O A096845 1,2
%A A096845 Labos E. (labos(AT)ana1.sote.hu), Jul 15 2004

%I A056661
%S A056661 0,1,2,5,8,11,29,31,182,296,491
%N A056661 Numbers n such that 40*R_n + 3 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A056661 Also numbers n such that (4*10^(n+1)-13)/9 is prime.
%t A056661 Do[ If[ PrimeQ[ 40*(10^n - 1)/9 + 3], Print[n]], {n, 0, 5000}]
%K A056661 hard,more,nonn
%O A056661 0,3
%A A056661 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 09 2000

Form 444..447
%I A099413
%S A099413 0,1,2,4,10,20,26,722,1310,3170
%N A099413 Numbers n such that 4*R_n + 3 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A099413 Also numbers n such that (4*10^n+23)/9 is prime.
%t A099413 Do[ If[ PrimeQ[ 4(10^n - 1)/9 + 3], Print[n]], {n, 5000}]
%Y A099413 Cf. .
%O A099413 0,3
%K A099413 nonn
%A A099413 Robert G. Wilson v (rgwv at rgwv.com), Oct 14 2004

%I A056682
%S A056682 0,1,3,9,19,25,721,1309,3169
%N A056682 Numbers n such that 40*R_n + 7 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A056682 Also numbers n such that (4*10^(n+1)+23)/9 is prime.
%t A056682 Do[ If[ PrimeQ[40*(10^n - 1)/9 + 7], Print[n]], {n, 0, 5000}]
%K A056682 hard,nonn
%O A056682 0,3
%A A056682 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 10 2000

Form 444..449
%I A099414
%S A099414 0,3,5,6,48,108,245,1044
%N A099414 Numbers n such that 4*R_n + 5 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A099414 Also numbers n such that (4*10^n+41)/9 is prime.
%t A099414 Do[ If[ PrimeQ[ 4(10^n - 1)/9 + 5], Print[n]], {n, 5000}]
%Y A099414 Cf. .
%O A099414 0,3
%K A099414 nonn
%A A099414 Robert G. Wilson v (rgwv at rgwv.com), Oct 14 2004

%I A056683
%S A056683 2,4,5,47,107,244,1043
%N A056683 Numbers n such that 40*R_n + 9 is prime, where R_n = 11...1 is the 
repunit (A002275).
%C A056683 Also numbers n such that (4*10^(n+1)+41)/9 is prime.
%t A056683 Do[ If[ PrimeQ[40*(10^n - 1)/9 + 9], Print[n]], {n, 5000}]
%K A056683 hard,nonn
%O A056683 0,1
%A A056683 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 10 2000

Form 555..551
%I A099415
%S A099415 12,13,609
%N A099415 Numbers n such that 5*R_n - 4 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A099415 Also numbers n such that (5*10^n-41)/9 is prime.
%t A099415 Do[ If[ PrimeQ[ 5(10^n - 1)/9 - 4], Print[n]], {n, 15000}]
%Y A099415 Cf. .
%O A099415 0,1
%K A099415 nonn
%A A099415 Robert G. Wilson v (rgwv at rgwv.com), Oct 14 2004

%I A056684
%S A056684 11,12,608
%N A056684 Numbers n such that 50*R_n + 1 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A056684 Also numbers n such that (5*10^(n+1)-41)/9 is prime.
%t A056684 Do[ If[ PrimeQ[50*(10^n - 1)/9 + 1], Print[n]], {n, 15000}]
%K A056684 hard,nonn,bref
%O A056684 0,1
%A A056684 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 10 2000

Form 555..553
%I A099416
%S A099416 0,1,2,8,26,66,74,233,473,540,2774
%N A099416 Numbers n such that 5*R_n - 2 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A099416 Also numbers n such that (5*10^n-23)/9 is prime.
%t A099416 Do[ If[ PrimeQ[ 5(10^n - 1)/9 - 2], Print[n]], {n, 0, 5000}]
%Y A099416 Cf. .
%O A099416 0,3
%K A099416 nonn
%A A099416 Robert G. Wilson v (rgwv at rgwv.com), Oct 14 2004

%I A056685
%S A056685 0,1,7,25,65,73,232,472,539,2773
%N A056685 Numbers n such that 50*R_n + 3 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A056685 Also numbers n such that (5*10^(n+1)-23)/9 is prime.
%t A056685 Do[ If[ PrimeQ[50*(10^n - 1)/9 + 3], Print[n]], {n, 0, 5000}]
%K A056685 hard,nonn
%O A056685 0,3
%A A056685 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 10 2000

Form 555..557
%I A099417
%S A099417 0,1,3,4,6,10,15,22,88,207,528,960,2100
%N A099417 Numbers n such that 5*R_n + 2 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A099417 Also numbers n such that (5*10^n+13)/9 is prime.
%t A099417 Do[ If[ PrimeQ[ 5(10^n - 1)/9 + 2], Print[n]], {n, 0, 5000}]
%Y A099417 Cf. .
%O A099417 0,3
%K A099417 nonn
%A A099417 Robert G. Wilson v (rgwv at rgwv.com), Oct 14 2004

%I A056686
%S A056686 0,2,3,5,9,14,21,87,206,527,959,2099
%N A056686 Numbers n such that 50*R_n + 7 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A056686 Also numbers n such that (5*10^(n+1)+13)/9 is prime.
%t A056686 Do[ If[ PrimeQ[50*(10^n - 1)/9 + 7], Print[n]], {n, 0, 5000}]
%K A056686 hard,nonn
%O A056686 0,2
%A A056686 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 10 2000

Form 555..559
%I A099418
%S A099418 2,8,12,18,26,32,138,188,222,338,1002,2744
%N A099418 Numbers n such that 5*R_n + 4 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A099418 Also numbers n such that (5*10^n+31)/9 is prime.
%t A099418 Do[ If[ PrimeQ[ 5(10^n - 1)/9 + 4], Print[n]], {n, 0, 5000}]
%Y A099418 Cf. .
%O A099418 0,3
%K A099418 nonn
%A A099418 Robert G. Wilson v (rgwv at rgwv.com), Oct 14 2004

%I A056687
%S A056687 1,7,11,17,25,31,137,187,221,337,1001,2743
%N A056687 Numbers n such that 50*R_n + 9 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A056687 Also numbers n such that (5*10^(n+1)+31)/9 is prime.
%t A056687 Do[ If[ PrimeQ[50*(10^n - 1)/9 + 9], Print[n]], {n, 0, 5000}]
%K A056687 hard,nonn
%O A056687 0,2
%A A056687 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 10 2000

Form 666..661
%I A098088
%S A098088 2,3,4,10,18,21,22,28,43,66,121,133,178,241,454,553,1600,2175,2978,3649
%N A098088 Numbers n such that 6*R_n - 5 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A098088 Also numbers n such that (2*10^n-17)/3 is prime.
%C A098088 No others less than 7000. n = 1600, 2175, 2978 and 3649 are only 
probable primes.
%H A098088 M. Kamada, <a 
href="http://homepage2.nifty.com/m_kamada/math/66661.htm">Factorizations of 
66...661</a>.
%e A098088 If n = 4 we get ((2*10^4)-17)/3 = 19983/3 = 6661, which is prime.
%t A098088 Do[ If[ PrimeQ[ 2(10^n - 1)/3 - 5], Print[n]], {n, 0, 7000}]
%K A098088 more,nonn
%O A098088 0,1
%A A098088 Julien Peter Benney (jpbenney(AT)ftml.net), Sep 14 2004

%I A056658
%S A056658 1,2,3,9,17,20,21,27,42,65,120,132,177,240,453,552,1599,2174,2977,3648
%N A056658 Numbers n such that 60*R_n + 1 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A056658 Also numbers n such that (2*10^(n+1)-17)/3 is prime.
%t A056658 Do[ If[ PrimeQ[ 60*(10^n - 1)/9 + 1], Print[n]], {n, 7000}]
%Y A056658 Cf. A092571 (corresponding primes).
%K A056658 hard,nonn
%O A056658 0,2
%A A056658 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 09 2000
%E A056658 1599 and 2174 (corresponding to probable primes) from Rick L. Shepherd 
(rshepherd2(AT)hotmail.com), Feb 28 2004

Form 666..667
%I A096507
%S A096507 1,2,6,8,9,11,20,23,41,63,66,119,122,149,252,284,305,592,746,875,1204,
%T A096507 1364,2240,2403,5106,5776,5813,12456,14235
%N A096507 Numbers n such that 6*R_n + 1 is a prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A096507 Also numbers n such that (2*10^n+1)/3 is prime.
%C A096507 All numbers from n = 2403 onwards are only probable primes. No others 
less than 25557. These numbers form a near-repdigit sequence (6)w7.
%e A096507 n = 9 gives 2000000001/3 = 666666667, which is prime.
%e A096507 n=20 means that 66666666666666666667 is a prime number.
%H A096507 M. Kamada, <a 
href="http://homepage2.nifty.com/m_kamada/math/66667.htm">Factorizations of 
66...667</a>.
%Y A096507 Cf. A096503, A096504, A096505, A096506, A096507, A096508.
%K A096507 base,nonn
%O A096507 0,2
%A A096507 Labos E. (labos(AT)ana1.sote.hu), Jul 12 2004
%E A096507 More terms from Julien Peter Benney (jpbenney(AT)ftml.net), Sep 14 2004

This is the same A098087 so simply delete A098087.

%I A056657
%S A056657 0,1,5,7,8,10,19,22,40,62,65,118,121,148,251,283,304,591,745,874,1203,
%T A056657 1363,2239,2402,5105,5775,5812,12455,14234
%N A056657 Numbers n such that 60*R_n + 7 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A056657 Also numbers n such that 2*(10^(n+1) - 1)/3 +1 is prime. 
???????????????????????????????????????????
%e A056657 7, 67, 666667, 66666667, 666666667, 66666666667, etc. are primes.
%t A056657 Do[ If[ PrimeQ[ 60*(10^n - 1)/9 + 7 ], Print[n]], {n, 25556}]
%K A056657 hard,nonn
%O A056657 0,3
%A A056657 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 09 2000
%E A056657 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Oct 22 2003
%E A056657 2239,2402,5105,5775 from Farideh Firoozbakht 
(f.firoozbakht(AT)sci.ui.ac.ir), Dec 23 2003

Form 777..771
%I A099419
%S A099419 2,13,20,23,31,100,241,275,925,1067,1369,2065
%N A099419 Numbers n such that 7*R_n - 6 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A099419 Also numbers n such that (7*10^n-61)/9 is prime.
%t A099419 Do[ If[ PrimeQ[ 7(10^n - 1)/9 - 6], Print[n]], {n, 0, 7000}]
%Y A099419 Cf. .
%O A099419 1,1
%K A099419 nonn
%A A099419 Robert G. Wilson v (rgwv at rgwv.com), Oct 14 2004

%I A056688
%S A056688 1,12,19,22,30,99,240,274,924,1066,1368,2064
%N A056688 Numbers n such that 70*R_n + 1 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A056688 Also numbers n such that (7*10^(n+1)-61)/9 is prime.
%t A056688 Do[ If[ PrimeQ[70*(10^n - 1)/9 + 1], Print[n]], {n, 0, 7000}]
%K A056688 hard,nonn
%O A056688 0,2
%A A056688 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 10 2000

Form 777..773
%I A099420
%S A099420 1,2,3,5,9,12,15,21,264,383,2720,4494
%N A099420 Numbers n such that 7*R_n - 4 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A099420 Also numbers n such that (7*10^n-43)/9 is prime.
%t A099420 Do[ If[ PrimeQ[ 7(10^n - 1)/9 - 4], Print[n]], {n, 0, 5000}]
%Y A099420 Cf. .
%O A099420 1,2
%K A099420 nonn
%A A099420 Robert G. Wilson v (rgwv at rgwv.com), Oct 14 2004

%I A056689
%S A056689 0,1,2,4,8,11,14,20,263,382,2719,4493
%N A056689 Numbers n such that 70*R_n + 3, where R_n = 11...1 is the repunit 
(A002275) of length n.
%C A056689 Also numbers n such that (7*10^(n+1)-43)/9 is prime.
%t A056689 Do[ If[ PrimeQ[70*(10^n - 1)/9 + 3], Print[n]], {n, 0, 5000}]
%K A056689 hard,nonn
%O A056689 0,3
%A A056689 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 10 2000

Form 777..779
%I A098089
%S A098089 2,66,86,90,102,386,624
%N A098089 Numbers n such that 7*R_n + 2 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A098089 Also numbers n such that (7*10^n+11)/9 is prime.
%C A098089 n = 386 and n = 624 are only probably prime. The next term is greater 
than 5,000 and is probably 7784.
%H A098089 M. Kamada, <a 
href="http://homepage2.nifty.com/m_kamada/math/77779.htm">Factorizations of 
77...779</a>.
%e A098089 If n = 2, we get ((7*10^2)+11/9 = (700+11)/9 = 79, which is prime.
%t A098089 Do[ If[ PrimeQ[ 7(10^n - 1)/9 + 2], Print[n]], {n, 5000}] (from RGWv 
Oct 15 2004)
%K A098089 more,nonn,new
%O A098089 0,1
%A A098089 Julien Peter Benney (jpbenney(AT)ftml.net), Sep 14 2004

%I A056693
%S A056693 1,65,85,89,101,385,623
%N A056693 Numbers n such that 70*R_n + 9 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A056693 Also numbers n such that (7*10^(n+1)+11)/9 is prime.
%t A056693 Do[ If[ PrimeQ[70*(10^n - 1)/9 + 9], Print[n]], {n, 0, 5000}]
%K A056693 hard,nonn
%O A056693 0,2
%A A056693 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 10 2000

Form 888..881
%I A099421
%S A099421 0,3,19,79,139,223,463,544,1096,1419,3247,3877,4417,9507,11091,14602
%N A099421 Numbers n such that 8*R_n - 7 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A099421 Also numbers n such that (8*10^n-71)/9 is prime.
%t A099421 Do[ If[ PrimeQ[ 8(10^n - 1)/9 - 7], Print[n]], {n, 0, 15000}]
%O A099421 0,2
%K A099421 nonn
%A A099421 Robert G. Wilson v (rgwv at rgwv.com), Oct 14 2004

%I A056664
%S A056664 2,18,78,138,222,462,543,1095,1418,3246,3876,4416,9506,11090,14601
%N A056664 Numbers n such that 80*R_n + 1 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A056664 Also numbers n such that (8*10^(n+1)-71)/9 is prime.
%t A056664 Do[ If[ PrimeQ[ 80*(10^n - 1)/9 + 1 ], Print[n]], {n, 15000}]
%Y A056664 Cf. A092675 (corresponding primes).
%K A056664 hard,nonn
%O A056664 0,1
%A A056664 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 09 2000
%E A056664 1415 (giving a probable prime) from Rick L. Shepherd 
(rshepherd2(AT)hotmail.com), Mar 02 2004. There are no other terms <= 2500.

Form 888..883
%I A099422
%S A099422 0,1,2,3,5,8,9,15,51,71,77,224,296,315,2090,2906,3395,3882
%N A099422 Numbers n such that 8*R_n - 5 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A099422 Also numbers n such that (8*10^n-53)/9 is prime.
%t A099422 Do[ If[ PrimeQ[ 8(10^n - 1)/9 - 5], Print[n]], {n, 0, 5000}]
%O A099422 0,3
%K A099422 nonn
%A A099422 Robert G. Wilson v (rgwv at rgwv.com), Oct 14 2004

%I A056694
%S A056694 0,1,2,4,7,8,14,50,70,76,223,295,314,2089,2905,3394,3881
%N A056694 Numbers n such that 80*R_n + 3 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A056694 Also numbers n such that (8*10^(n+1)-53)/9 is prime.
%t A056694 Do[ If[ PrimeQ[80*(10^n - 1)/9 + 3], Print[n]], {n, 0, 5000}]
%K A056694 hard,nonn
%O A056694 0,3
%A A056694 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 10 2000

Form 888..887
%I A096846
%S A096846 1,3,4,6,9,12,72,118,124,190,244,304,357,1422,2691
%N A096846 Numbers n for which -1+8*R_n - 1 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A096846 Also numbers n such that (8*10^n-17)/9 is prime.
%e A096846 n=72: 
888888888888888888888888888888888888888888888888888888888888888888888887 is prime.
%t A096846 Do[ If[ PrimeQ[ 8(10^n - 1)/9 + 1], Print[n]], {n, 5000}] (from RGWv 
Oct 15 2004)
%Y A096846 Cf. A096503, A096504, A096505, A096506, A096507, A096508, A096841, 
A096842, A096843, A096844, A096845, A096846, A000203.
%K A096846 more,nonn
%O A096846 1,2
%A A096846 Labos E. (labos(AT)ana1.sote.hu), Jul 15 2004

%I A056695
%S A056695 0,2,3,5,8,11,71,117,123,189,243,303,356,1421,2690
%N A056695 Numbers n such that 80*R_n + 7 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A056695 Also numbers n such that (8*10^(n+1)-17)/9 is prime.
%t A056695 Do[ If[ PrimeQ[80*(10^n - 1)/9 + 7], Print[n]], {n, 0, 5000}]
%K A056695 hard,nonn
%O A056695 0,2
%A A056695 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 10 2000

Form 888..889
%I A096508
%S A096508 2,14,17,35,4175,4472,14576
%N A096508 Numbers n for which 8*R_n + 1 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A096508 Also numbers n such that (8*10^n+1)/9 is prime.
%e A096508 n=35 means that 88888888888888888888888888888888889 is a prime number.
%t A096508 Do[ If[ PrimeQ[ 8(10^n - 1)/9 + 1], Print[n]], {n, 30000}] (from RGWv 
Oct 15 2004)
%Y A096508 Cf. A096503, A096504, A096505, A096506, A096507.
%K A096508 base,nonn
%O A096508 1,1
%A A096508 Labos E. (labos(AT)ana1.sote.hu), Jul 12 2004

%I A056663
%S A056663 1,13,16,34,4174,4471,14575
%N A056663 Numbers n such that 80*R_n + 9 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A056663 Also numbers n such that (8*10^(n+1)+1)/9 is prime.
%t A056663 Do[ If[ PrimeQ[ 80*(10^n - 1)/9 + 9 ], Print[n]], {n, 0, 30000}]
%K A056663 hard,nonn
%O A056663 0,2
%A A056663 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 09 2000

Form 999..991
%I A095714
%S A095714 3,5,7,33,45,105,197,199,281,301,317,1107,1657,3395
%N A095714 Numbers n such that 90*R_n + 1 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A095714 Also numbers n such that 10^n - 9 is a prime.
%C A095714 No more terms below 4300. RGWv Oct 15 2004.
%e A095714 a(2) = 5, since 10^5 - 9 = 99991, which is prime
%t A095714 Do[ If[ PrimeQ[10^n - 9], Print[n]], {n, 0, 7000}]
%Y A095714 Cf. A088275.
%K A095714 nonn
%O A095714 0,1
%A A095714 Alonso Delarte (alonso.delarte(AT)gmail.com), Jul 07 2004
%E A095714 More terms from RGWv (rgwv at rgwv.com), Oct 15 2004

%I A056696
%S A056696 2,4,6,32,44,104,196,198,280,300,316,1106,1656,3394
%N A056696 Numbers n such that 90*R_n + 1 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A056696 Also numbers n such that 10^(n+1) - 9 is a prime.
%t A056696 Do[ If[ PrimeQ[90*(10^n - 1)/9 + 1], Print[n]], {n, 0, 7000}]
%K A056696 hard,nonn
%O A056696 0,1
%A A056696 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 10 2000

Form 999..997
%I A089675
%S A089675 1,2,3,17,140,990,1887,3530,5996,13820,21873
%N A089675 Numbers n such that 9*R_n - 2 is a prime number, where R_n = 11...1 is 
the repunit (A002275) of length n.
%C A089675 Also numbers n such that 10^n - 3 is a prime number.
%C A089675 Next term is greater than 22500. - Gabriel Cunningham 
(gcasey(AT)mit.edu), Mar 13 2004
%C A089675 26045 is a known prime. Henri Lifchitz.
%e A089675 10^2 - 3 = 97 is a prime number (in fact the largest less than 10^2).
%t A089675 To check for all n up to m: 
For[n=1,n<m,If[PrimeQ[10^n-3]==True,Print[n]];n++ ]
%K A089675 more,nonn
%O A089675 1,2
%A A089675 Michael Gottlieb (mzrg(AT)verizon.net), Jan 05 2004
%E A089675 a(8) from Robert G. Wilson v, Jan 14 2004.
%E A089675 a(9) and a(10) from Gabriel Cunningham (gcasey(AT)mit.edu), Mar 06 2004
%E A089675 a(11) from Gabriel Cunningham (gcasey(AT)mit.edu), Mar 13 2004

%I A056662
%S A056662 0,1,2,16,139,989,1886,3529,5995,13819,21872
%N A056662 Numbers n such that 90*R_n + 7 is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A056662 Also numbers n such that 10^(n+1) - 3 is a prime number.
%t A056662 Do[ If[ PrimeQ[ 90*(10^n - 1)/9 + 7 ], Print[n]], {n, 0, 1250}]
%K A056662 hard,nonn
%O A056662 0,3
%A A056662 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 09 2000