cp's OEIS Frontend

Equivalently, numbers k for which there is at least one j such that 2^j * 3^(k-j) is the average of a twin prime pair.

The only even term is 2: the corresponding twin prime pairs are 2^2 * 3^0 -+ 1 = (3,5) and 2^1 * 3^1 -+ 1 = (5,7), each of which includes 5 as an element of the pair. If k is even, 2^j * 3^(k-j) differs by 1 from a multiple of 5 for every j.

The exponent, k, of 2 must be odd because the exponent, 2, of 3 (where 9 = 3^2) is even and the sum of the exponents of prime factors 2 and 3 must be odd to form a product that is a twin prime average. Of all subsequences of A027856, this is the longest known where the power of 3 is fixed.

Amiram Eldar noted that using A002236 and A002256, we obtain a(7) > 5.6*10^6, if it exists.

a(28) > 10^5.

Conjecture: This sequence intersects with A387201 at k = 4 to form twin primes with center N = 2^5 * 3^4 = 2592 = A027856(10). Any such intersection has to be at an even k because if k is odd, either N-1 or N+1 has to be divisible by 5. A covering system can be constructed that eliminates all other intersections except where k = 4(mod 60), and for k > 4 with k = 4(mod 60), the search up to 10^5 makes the probability of another intersection in this residue class vanishingly small.

a(23) > 10^5.

Conjecture: This sequence intersects with A387197 at k = 4 to form twin primes with center N = 2^5 * 3^4 = 2592 = A027856(10). Any such intersection has to be at an even k because if k is odd, either N-1 or N+1 has to be divisible by 5. A covering system can be constructed that eliminates all other intersections except where k == 4 (mod 60), and for k > 4 with k == 4 (mod 60), the search up to 10^5 makes the probability of another intersection in this residue class vanishingly small.