A086082 Numbers m such that m and all of its even complements from 2 to 10 are primes. In other words, m and j^k - m (where k is the smallest power of j such that j^k > m) are prime for all of the following values of j: 2, 4, 6, 8, 10.

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

Offset: 1

Author

Chuck Seggelin, Jul 08 2003

Keywords

• nonn

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 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 raised to any power > 0.

Examples

			887 is a term 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.

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)