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A118584
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Least n-digit prime in base 12 that is a member of twin prime pair and also either a Sophie Germain prime or a safe prime.
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0
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3, 29, 179, 1931, 20771, 248867, 2986349, 35833079, 429983039, 5159780471, 61917366011, 743008372451, 8916100451231, 106993205386139, 1283918464561721, 15407021574604589, 184884258895077527, 2218611106740456359, 26623333280885245319, 319479999370622955167, 3833759992447475131451
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OFFSET
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1,1
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COMMENTS
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Let X be 10 and E be 11 in base 12. In base 12, all primes greater than 3 end in the digits 1,5,7,E. All twin primes (p,q) with p>3 end in the digits (5,7) or (E,1). All Sophie Germain primes of type 1 end in the digit 5 or E. A Cunningham chain of type 1 starts with a 5-prime or E-prime and all subsequent primes are E-primes. The sequence in base 12 is "3", "25", "12E", "114E", "1002E", "10002E", "1000265", "1000089E", "10000093E", "100000009E", "1000000104E", "10000000102E", "100000000187E", "1000000000410E", "100000000007535", "100000000000X665", "1000000000001E95E", "10000000000000E25E", "100000000000000099E", "1000000000000001447E", "10000000000000000544E", "100000000000000002797E", "1000000000000000000872E", "10000000000000000006806E".
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LINKS
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EXAMPLE
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29 is 25 in base 12 and is the first two digit prime that is twin prime (31 = 27 is its companion) and Sophie Germain prime, since 2*29+1 = 59 = 4E is prime.
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MAPLE
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istwin := proc(p::prime) isprime(p-2) or isprime(p+2) end: issophie := proc(p::prime) isprime(2*p+1) or isprime((p-1)/2) end: L:=[]: for w to 1 do for n from 1 to 24 do p:=nextprime(12^(n-1)); while not (istwin(p) and issophie(p)) do p:=nextprime(p) od; L:=[op(L), p]; od; od; L;
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MATHEMATICA
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a[n_] := Module[{p = NextPrime[12^(n-1)]}, While[!Or @@ PrimeQ[p + {-2, 2}] || !Or @@ PrimeQ[{(p-1)/2, 2*p+1}], p = NextPrime[p]]; p]; Array[a, 25] (* Amiram Eldar, Aug 04 2024 *)
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CROSSREFS
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KEYWORD
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easy,nonn,base
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AUTHOR
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EXTENSIONS
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Name corrected and a(18)-a(21) added by Amiram Eldar, Aug 04 2024
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STATUS
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approved
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