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

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

(PARI) forprime(p=1, 1000, k=0; x=3; y=1; while(k<p+1, while(x<y+p, x=3*x); if(x-y==p, print1(p, ", "); k=p); k++; y=2*y))

Crossrefs

Cf. A000040, A192111.

Cf. A007643 (Primes not of form |3^x - 2^y|).

Cf. A321671 (Primes of the form 2^j - 3^k).

Sequence in context: A235480 A049042 A020605 * A045348 A023207 A038611

Adjacent sequences: A323695 A323696 A323697 * A323699 A323700 A323701

Keyword

nonn

Author

Jinyuan Wang, Jan 24 2019

Status

approved