A360993 Numbers k such that (2^k - 1)^3 + 2 is a semiprime.

4, 5, 8, 12, 13, 18, 20, 29, 38, 56, 60, 62, 76, 82, 101, 118, 202, 210, 230, 276, 328, 332, 336, 338, 368

Offset: 1

Author

Serge Batalov, Feb 27 2023

Keywords

• nonn
• more
• hard

Comments

a(26) >= 406.

438, 500, 526, 604, 648, 696 are also in this sequence, but their positions cannot be established before finding any factor for the values corresponding to the following "blockers": 406, 496, 528.

2382, 2733, 2910, 3368, 3508, 5338, 7705, 11185, 19905, 23814, 38545, 179294 are larger terms of this sequence, but their positions cannot be established. These produce "trivial" semiprimes where one prime is small (e.g., 3 or 11).

Examples

			a(1) = 4 because 15^3 + 2 = 3377 = 11 * 307, which is semiprime.
a(2) = 5 because 31^3 + 2 = 29793 = 3 * 9931, which is semiprime.
		

Links

• factordb.com, Status of (2^406-1)^3+2.

Crossrefs

Cf. A091515, A091516, A100899, A100900, A268574, A269264, A360994.

Programs

Magma

IsSemiprime:=func; [n: n in [2..70]| IsSemiprime(s) where s is (2^n-1)^3+2];

Mathematica

Select[Range[70], PrimeOmega[(2^# - 1)^3 - 2] == 2 &]

PARI

isok(n) = bigomega((2^n-1)^3+2) == 2;

Extensions

a(20)-a(26) from Serge Batalov, Mar 03 2023