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

1, 3, 21, 57, 219, 399, 903, 1533, 2667, 4161, 7077, 17157, 18543, 29127, 32547, 50673, 65919, 74109, 96141, 113799, 114681, 134463, 194691, 227829, 304311, 352317, 383907, 389193, 463071, 516621, 672987, 797349, 863517, 898779, 932799, 1252461, 1353639

Offset: 1

Author

Alexander Adamchuk, Apr 03 2007

Keywords

• nonn

Comments

From Robert Israel, Aug 13 2020: (Start)

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

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

Are all terms squarefree? (End)

Links

• Robert Israel, Table of n, a(n) for n = 1..500 (first 49 terms from Robert Price)

Crossrefs

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

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

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

Programs

Maple

filter:= n -> 4&^(n^2)-1 mod (n^3) = 0:
select(filter, [1,seq(i,i=3..10^6,6)]); # Robert Israel, Aug 13 2020

Mathematica

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

Extensions

More terms from Robert G. Wilson v, Apr 06 2007