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A129212 Numbers k such that k^3 divides 4^(k^2) - 1.

%I A129212 #17 Jun 08 2021 02:23:42
%S A129212 1,3,21,57,219,399,903,1533,2667,4161,7077,17157,18543,29127,32547,
%T A129212 50673,65919,74109,96141,113799,114681,134463,194691,227829,304311,
%U A129212 352317,383907,389193,463071,516621,672987,797349,863517,898779,932799,1252461,1353639
%N A129212 Numbers k such that k^3 divides 4^(k^2) - 1.
%C A129212 From _Robert Israel_, Aug 13 2020: (Start)
%C A129212 Except for 1, all terms are divisible by 3, but not 5 or 9.
%C A129212 All terms > 3 are divisible by at least one of 7, 19 and 73.
%C A129212 Are all terms squarefree? (End)
%H A129212 Robert Israel, <a href="/A129212/b129212.txt">Table of n, a(n) for n = 1..500</a> (first 49 terms from Robert Price)
%p A129212 filter:= n -> 4&^(n^2)-1 mod (n^3) = 0:
%p A129212 select(filter, [1,seq(i,i=3..10^6,6)]); # _Robert Israel_, Aug 13 2020
%t A129212 k=2; Do[ p=Prime[k]; If[ IntegerQ[ (PowerMod[ p+1, n^2, n^3 ] - 1 )/n^3 ], Print[ {k, p, n} ]], {n,1,200000} ]
%t A129212 k=2; Do[ p=Prime[k]; If[ IntegerQ[ (PowerMod[ p+1, n^2, n^3 ] - 1 )/n^3 ], Print[ {k, p, n} ]], {n,1000000} ] (* _Robert G. Wilson v_, Apr 06 2007 *)
%t A129212 Join[{1}, Select[Range[3000000], PowerMod[4, #^2, #^3] == 1 &]] (* _Robert Price_, Mar 31 2020 *)
%Y A129212 Cf. A014945 (numbers k such that k divides 4^k-1).
%Y A129212 Cf. A127104 (numbers k such that k^2 divides 4^k-1).
%Y A129212 Cf. A128678 (numbers k such that k^3 divides 4^(k^2)+1).
%K A129212 nonn
%O A129212 1,2
%A A129212 _Alexander Adamchuk_, Apr 03 2007
%E A129212 More terms from _Robert G. Wilson v_, Apr 06 2007