A309738
Primes of the form b^2*10^(2*k) + b*10^k + 1 for 1 <= b <= 9, k >= 0.
Original entry on oeis.org
3, 7, 13, 31, 43, 73, 421, 2551, 6481, 8191, 250501, 4002001, 64008001, 81009001, 40000200001, 90000300001, 64000008000001, 400000000020000000001, 3600000000060000000001, 64000000000008000000000001, 90000000000000300000000000001, 250000000000000500000000000001
Offset: 1
b | Primes of the form b^2*10^(2*k) + b*10^k + 1
--+-------------------------------------------------------------
1 | 3.
2 | 7, 421, 4002001, 40000200001, 400000000020000000001, ...
3 | 13, 90000300001, 90000000000000300000000000001, ...
4 |
5 | 31, 2551, 250501, 250000000000000500000000000001, ...
6 | 43, 3600000000060000000001, ...
7 |
8 | 73, 6481, 64008001, 64000008000001, ...
9 | 8191, 81009001, 810000000000000000900000000000000001, ...
A296444
Numbers k such that 2*10^(2k) + 2*10^k + 1 are prime.
Original entry on oeis.org
0, 2, 3, 6, 10, 276, 746, 1090, 1485, 6186, 8571, 15594
Offset: 1
5, 20201, 2002001, 2000002000001, and 200000000020000000001 are prime, while 221=13*17, 200020001=569*351529, and 20000200001=17*29*1129*35933.
See
A296443 for 2*10^(2k)-2*10^k+1.
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ParallelMap[ If[ PrimeQ[2*10^(2 #) + 2*10^# + 1], #, Nothing] &, Range@ 6500] (* Robert G. Wilson v, Dec 13 2017 *)
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isok(k) = isprime(2*10^(2*k)+2*10^k+1); \\ Michel Marcus, Dec 13 2017
A309743
Numbers k such that 9*10^(2*k) + 9*10^k + 1 is prime.
Original entry on oeis.org
0, 1, 2, 6, 12, 245, 298, 2967, 4321, 7225, 11267
Offset: 1
19 is prime. ==> a(1) = 0.
991 is prime. ==> a(2) = 1.
90901 is prime. ==> a(3) = 2.
9009001 = 131 * 68771.
900090001 = 421 * 2137981.
90000900001 = 131 * 701 * 980071.
9000009000001 is prime. ==> a(4) = 6.
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for(k=0, 1e3, if(ispseudoprime(9*100^k+9*10^k+1), print1(k", ")))
A309742
Numbers k such that 8*10^(2*k) + 8*10^k + 1 are prime.
Original entry on oeis.org
0, 1, 6, 11, 23, 297, 474, 1121, 2531, 3573, 5437, 5919
Offset: 1
17 is prime. ==> a(1) = 0.
881 is prime. ==> a(2) = 1.
80801 = 7^2 * 17 * 97.
8008001 = 47 * 170383.
800080001 = 7 * 23 * 103 * 48247.
80000800001 = 71 * 1126771831.
8000008000001 is prime. ==> a(3) = 6.
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for(k=0, 1e3, if(ispseudoprime(8*100^k+8*10^k+1), print1(k", ")))
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from sympy import isprime
def afind(limit, startk=0):
for k in range(startk, limit+1):
if isprime(8*100**k + 8*10**k + 1): print(k, end=", ")
afind(500) # Michael S. Branicky, Dec 12 2021
A309740
Numbers k such that 5*10^(2*k) + 5*10^k + 1 is prime.
Original entry on oeis.org
0, 3, 5, 301, 13817, 15259
Offset: 1
11 is prime ==> a(1) = 0.
551 = 19 * 29.
50501 = 11 * 4591.
5005001 is prime ==> a(2) = 3.
500050001 = 11 * 61 * 745231.
50000500001 is prime ==> a(3) = 5.
5000005000001 = 11 * 31 * 1801 * 8141461.
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for(k=0, 1e3, if(ispseudoprime(5*100^k+5*10^k+1), print1(k", ")))
A309741
Numbers k such that 6*10^(2*k) + 6*10^k + 1 is prime.
Original entry on oeis.org
0, 1, 2, 4, 22, 133, 567, 14739, 25390
Offset: 1
13 is prime. ==> a(1) = 0.
661 is prime. ==> a(2) = 1.
60601 is prime. ==> a(3) = 2.
6006001 = 2027 * 2963.
600060001 is prime. ==> a(4) = 4.
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for(k=0, 1e3, if(ispseudoprime(6*100^k+6*10^k+1), print1(k", ")))
Showing 1-6 of 6 results.
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