1,1

Primes meeting the requirements to be members of this sequence are fairly rare. The 653th prime in this sequence is the 672448th prime in the sequence of all primes (i.e. 0.0971% of the first 672448 primes belong in this sequence.) Primes which need only be j-complement for one value of j (such as 6-complement primes) are relatively common (in the first 672509 primes, 122932 are 6-complement 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, odd-complement primes are either 2 or of the form j^k-2 - for example, the first few 7's complement primes are 2, 5 (7^1-2), 47 (7^2-2), 2399 (7^4-2), 823541 (7^7-2), 5764799 (7^8-2), 13841287199 (7^12-2), 4747561509941 (7^15-2), 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.

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)

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.

Cf. A068811, A086081.

Sequence in context: A059472 A079593 A045711 this_sequence A126665 A107160 A075587

Adjacent sequences: A086079 A086080 A086081 this_sequence A086083 A086084 A086085

nonn

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