Least k such that 2^n - k - 2 and 2^n - k are twin primes.

1, 3, 1, 3, 19, 15, 49, 3, 19, 3, 103, 21, 49, 15, 61, 33, 67, 3, 19, 501, 157, 75, 421, 195, 289, 447, 73, 105, 697, 1455, 301, 375, 1069, 345, 199, 315, 529, 1377, 73, 1365, 799, 117, 511, 885, 1627, 525, 1993, 651, 787, 633, 2449, 255, 169, 2127, 1603, 1965

3

Distance from 2^n to the greater member of the greatest pair of twin primes less than 2^n.

T. D. Noe, Plot of terms 3 to 1000

T. D. Noe, Table of terms 3 to 1000

(Mma) Table[k = 1; While[! (PrimeQ[2^n - k] && PrimeQ[2^n - k - 2]), k = k + 2]; k, {n, 3, 50}]

Cf. A173937 (k such that 2^n+k and 2^n+k+2 are the first twin primes greater than 2^n), S000267.

nonn

T. D. Noe, Sep 22 2014