login
A118584
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.
0
3, 29, 179, 1931, 20771, 248867, 2986349, 35833079, 429983039, 5159780471, 61917366011, 743008372451, 8916100451231, 106993205386139, 1283918464561721, 15407021574604589, 184884258895077527, 2218611106740456359, 26623333280885245319, 319479999370622955167, 3833759992447475131451
OFFSET
1,1
COMMENTS
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".
EXAMPLE
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.
MAPLE
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;
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
easy,nonn,base
AUTHOR
Walter Kehowski, May 17 2006
EXTENSIONS
Name corrected and a(18)-a(21) added by Amiram Eldar, Aug 04 2024
STATUS
approved