cp's OEIS Frontend

A330382 Composite numbers k such that k-1 divides 2^k-2.

%I A330382 #30 Jul 21 2024 22:11:39
%S A330382 55,295,343,1027,1135,1315,1807,2059,2395,3403,4375,5335,6175,6499,
%T A330382 7183,7939,9235,10207,12643,13123,14155,16003,16255,19495,21547,23815,
%U A330382 27595,27703,30619,35479,37927,43219,45487,48007,48763,50275,55567,58483,64387,64639,74899
%N A330382 Composite numbers k such that k-1 divides 2^k-2.
%C A330382 If k is in this sequence, then 2^k-1 is also a term, so this sequence is infinite.
%C A330382 Also 2^p-1 is in this sequence for such prime p in A069051 that 2^p-1 is composite.
%C A330382 Theorem: if k-1 | 2^k-2, then m-1 | 2^m-2, where m = 2^k-1.
%C A330382 Conjecture: k-1 | 2^k-2 for k = (2^n-1)^3 if and only if n(n-1) | 2^n-2 for n > 2.
%C A330382 It seems that A007013(n)^3 for n > 1 and A007013(n) for n > 4 are in this sequence.
%C A330382 These are the composites k for which M - 1 divides 2^M - 2 where M = 2^k - 1. - _Thomas Ordowski_, Jul 01 2024
%H A330382 Amiram Eldar, <a href="/A330382/b330382.txt">Table of n, a(n) for n = 1..5000</a>
%F A330382 Composites of A014741(n) + 1. - _Thomas Ordowski_, Jul 01 2024
%t A330382 Select[Range[75000], CompositeQ[#] && Divisible[PowerMod[2, #, # - 1] - 2, # - 1] &]
%o A330382 (PARI) forcomposite(k=1,75000,if(!((2^k-2)%(k-1)),print1(k,", "))) \\ _Hugo Pfoertner_, Dec 12 2019
%Y A330382 Cf. A007013, A014741, A054723, A069051, A217468.
%Y A330382 A217468 is a subsequence.
%K A330382 nonn
%O A330382 1,1
%A A330382 _Amiram Eldar_ and _Thomas Ordowski_, Dec 12 2019