A323698 - OEIS

A323698

Primes of the form 3^j - 2^k, for j>=0, k>=0.

2, 5, 7, 11, 17, 19, 23, 73, 79, 139, 179, 211, 227, 239, 241, 601, 727, 1163, 1931, 2179, 3299, 4513, 6529, 6553, 11491, 19427, 19681, 50857, 58537, 58921, 111611, 144379, 176123, 177019, 177131, 529393, 545747, 1593299, 1594259, 2685817, 4782961, 9492289, 14346859

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OFFSET

1,1

COMMENTS

In this sequence, only 5 and 17 make both j and k even numbers.

Generally, the way to prove that a number is not in this sequence is to successively take residues modulo 3, 8, 5, and 16 on both sides of the equation 3^j - 2^k = x.

LINKS

Table of n, a(n) for n=1..43.

FORMULA

Intersection of A000040 and A192111.

EXAMPLE

11 = 3^3 - 2^4, so 11 is a term.

41 == 1 (mod 8), 41 == 2 (mod 3), so j = 2*l, k = 2*m. 41 == 1 (mod 5), but 3^(2*l) - 2^(2*m) mod 5 is 0, 2 or 3. So 41 is not in this sequence.

MATHEMATICA

c = 3; d = 2; t[i_, j_] := c^i - d^j;

u = Table[If[PrimeQ[t[i, j]] == True, u = t[i, j]], {i, 0, 20}, {j, 0, i*Log[d, c]}];

v = Union[Flatten[u]]

PROG