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Numbers k such that k^2 divides 4^k-1.
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%I #21 May 25 2024 09:10:44

%S 1,3,21,903,2667,7077,113799,114681,304311,389193,898779,932799,

%T 4893357,6099429,8131683,8776257,14452473,38350263,38647497,40647747,

%U 49427511,99583113,118465473,128794323,131158041,152643813,262275447,300510651,314353263,335873559,349662369

%N Numbers k such that k^2 divides 4^k-1.

%C From _Alexander Adamchuk_, Jan 11 2007: (Start)

%C 3 divides a(n) for n > 1.

%C 7 divides a(n) for n > 2.

%C 43 divides a(n) for n = {4, 8, 9, 10, 12, 13, 16, ...}.

%C 127 divides a(n) for n = {5, 8, 11, 14, 15, 17, ...}.

%C Prime factors of a(n) in order of their appearance in {a(n)} are {3, 7, 43, 127, 337, 5419, 431, 1033, 5419, 2287, 3049, 9719, ...}. (End)

%H Amiram Eldar, <a href="/A127104/b127104.txt">Table of n, a(n) for n = 1..56</a>

%t Select[Range[30000], IntegerQ[(PowerMod[4, #, #^2 ]-1)/#^2 ]&]

%o (PARI) is(k) = Mod(4, k^2)^k == 1; \\ _Amiram Eldar_, May 25 2024

%Y Cf. A127100, A127101, A127102, A127103, A127105, A127106, A127107, A127092.

%Y Subset of A014945 (numbers k such that k divides 4^nk-1).

%K nonn

%O 1,2

%A _Alexander Adamchuk_, Jan 05 2007

%E More terms from _Ryan Propper_ and _Alexander Adamchuk_, Jan 05 2007

%E a(27)-a(31) from _Amiram Eldar_, May 25 2024