

A086082


Numbers n such that n and all of its even complements from 2 to 10 are primes. In other words, n and j^k  n (where k is the smallest power of j such that j^k > n) are prime for all of the following values of j: 2, 4, 6, 8, 10.


0



53, 59, 557, 773, 887, 2207, 2273, 2543, 2789, 3209, 3449, 3677, 33347, 33893, 36887, 41927, 54323, 61547, 131303, 131687, 136217, 138143, 139493, 140177, 150083, 150533, 153353, 153437, 154277, 157007, 158303, 161333, 162263, 163847, 166157
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OFFSET

1,1


COMMENTS

Primes meeting the requirements to be members of this sequence are fairly rare. The 653rd prime in this sequence is the 672448th prime in the sequence of all primes (i.e., 0.0971% of the first 672448 primes belong to this sequence). Primes which need only be jcomplement for one value of j (such as 6complement primes) are relatively common (in the first 672509 primes, 122932 are 6complement primes, or about 18.28%).
Odd complement primes are very rare, simply because any odd number raised to a power yields an odd number. Subtracting from this an odd prime yields an even number that cannot be prime unless it is 2. As a result, oddcomplement primes are either 2 or of the form j^k2  for example, the first few 7's complement primes are 2, 5 (7^12), 47 (7^22), 2399 (7^42), 823541 (7^72), 5764799 (7^82), 13841287199 (7^122), 4747561509941 (7^152) and so forth. This is a natural result of the fact that most primes are odd and so are odd numbers when r aised to any power > 0.


LINKS

Table of n, a(n) for n=1..35.


FORMULA

If isPrime(p) And isPrime(2^(floor(Log(p, 2))+1)p) And isPrime(4^(floor(Log(p, 4))+1)p) And isPrime(6^(floor(Log(p, 6))+1)p) And isPrime(8^(floor(Log(p, 8))+1)p) And isPrime(10^(floor(Log(p, 10))+1)p) then sequence.add(p)


EXAMPLE

a(5)=887 because i: 887 is prime. ii: (2^10  887) = (1024  887) = 137 which is prime. iii: (4^5  887) = (1024  887) = 137 which is prime. iv: (6^4  887) = (1296  887) = 409 which is prime. v: (8^4  887) = (4096  887) = 3209 which is prime. vi: (10^3  887) = (1000  887) = 113 which is prime.


CROSSREFS

Cf. A068811, A086081.
Sequence in context: A165455 A180553 A079593 * A126665 A279191 A107160
Adjacent sequences: A086079 A086080 A086081 * A086083 A086084 A086085


KEYWORD

nonn


AUTHOR

Chuck Seggelin (barkeep(AT)plastereddragon.com), Jul 08 2003


STATUS

approved



