A360993 - OEIS

%I #30 Mar 03 2023 13:56:07

%S 4,5,8,12,13,18,20,29,38,56,60,62,76,82,101,118,202,210,230,276,328,

%T 332,336,338,368

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

%C a(26) >= 406.

%C 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.

%C 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).

%H factordb.com, <a href="http://factordb.com/index.php?query=%282%5E406-1%29%5E3%2B2">Status of (2^406-1)^3+2</a>.

%e a(1) = 4 because 15^3 + 2 = 3377 = 11 * 307, which is semiprime.

%e a(2) = 5 because 31^3 + 2 = 29793 = 3 * 9931, which is semiprime.

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

%o (Magma) IsSemiprime:=func<i | &+[d[2]: d in Factorization(i)] eq 2>; [n: n in [2..70]| IsSemiprime(s) where s is (2^n-1)^3+2];

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

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

%K nonn,more,hard

%O 1,1

%A _Serge Batalov_, Feb 27 2023

%E a(20)-a(26) from _Serge Batalov_, Mar 03 2023