A364699 Numbers k such that 1 + 2^k + 3^k is divisible by 2*k-1.

1, 4, 9, 16, 21, 40, 45, 52, 57, 64, 69, 76, 100, 112, 117, 129, 136, 141, 177, 184, 201, 220, 225, 232, 244, 261, 285, 297, 304, 309, 316, 321, 364, 376, 381, 405, 412, 429, 441, 460, 465, 477, 484, 489, 496, 520, 525, 532, 544, 549, 597, 609, 616, 640, 645, 652, 664, 681, 700, 705, 712, 717

Offset: 1

Author

Robert Israel, Aug 02 2023

Keywords

• nonn

Comments

k is a term if 2^(k-1) == 1 and 3^(k-1) == -1 (mod 2*k-1). This is true if 2*k-1 is in A290402. Terms such that 2*k-1 is not in A290402 include 1, 2586801, 3393664, and 7989004. Are there any terms such that 2^(k-1) and 3^(k-1) are not == 1 and -1 (respectively) mod 2*k-1?

Examples

			a(3) = 9 is a term because 1 + 2^9 + 3^9 = 20196 = 1188 * 17 is divisible by 2*9-1 = 17.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001550, A290402, A364683.

Programs

Maple

select(k -> 1 + 2 &^ k + 3 &^ k mod (2*k-1) = 0, [$1..1000]);

Mathematica

Select[Range[720], Divisible[PowerMod[3, #, 2*# - 1] + PowerMod[2, #, 2*# - 1] + 1, 2*# - 1] &] (* Amiram Eldar, Aug 03 2023 *)