A129212 - OEIS

%I #17 Jun 08 2021 02:23:42

%S 1,3,21,57,219,399,903,1533,2667,4161,7077,17157,18543,29127,32547,

%T 50673,65919,74109,96141,113799,114681,134463,194691,227829,304311,

%U 352317,383907,389193,463071,516621,672987,797349,863517,898779,932799,1252461,1353639

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

%C From _Robert Israel_, Aug 13 2020: (Start)

%C Except for 1, all terms are divisible by 3, but not 5 or 9.

%C All terms > 3 are divisible by at least one of 7, 19 and 73.

%C Are all terms squarefree? (End)

%H Robert Israel, <a href="/A129212/b129212.txt">Table of n, a(n) for n = 1..500</a> (first 49 terms from Robert Price)

%p filter:= n -> 4&^(n^2)-1 mod (n^3) = 0:

%p select(filter, [1,seq(i,i=3..10^6,6)]); # _Robert Israel_, Aug 13 2020

%t 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 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 Join[{1}, Select[Range[3000000], PowerMod[4, #^2, #^3] == 1 &]] (* _Robert Price_, Mar 31 2020 *)

%Y Cf. A014945 (numbers k such that k divides 4^k-1).

%Y Cf. A127104 (numbers k such that k^2 divides 4^k-1).

%Y Cf. A128678 (numbers k such that k^3 divides 4^(k^2)+1).

%K nonn

%O 1,2

%A _Alexander Adamchuk_, Apr 03 2007

%E More terms from _Robert G. Wilson v_, Apr 06 2007