A127103 Numbers k such that k^2 divides 3^k-1.

1, 2, 4, 20, 220, 1220, 2420, 5060, 13420, 14740, 23620, 55660, 145420, 147620, 162140, 237820, 259820, 290620, 308660, 339020, 447740, 847220, 899140, 1210220, 1440820, 1599620, 1759340, 2332660, 2616020, 2858020, 3196820, 3344660

Offset: 1

Author

Alexander Adamchuk, Jan 05 2007

Keywords

• nonn

Comments

From Alexander Adamchuk, Jan 11 2007: (Start)

2 divides a(n) for n>1. 2^2 divides a(n) for n>2. 5 divides a(n) for n>3.

11 divides a(n) for n = {5,7,8,9,10,12,13,14,15,16,17,18,19,20,22,23,24,26,27, 28,29,30,31,31,33,34,35,...}.

11^2 divides a(n) for n = {7,12,14,15,26,27,29,30,31,33,34,...}.

Prime factors of a(n) in order of their appearance in a(n) are {2,5,11,61,23,67,1181,661,47,1321,367,3851,5501,727,461,269,...}. (End)

Links

• Amiram Eldar, Table of n, a(n) for n = 1..300

Crossrefs

Subset of A067945 (numbers k that divide 3^k - 1).

Cf. A127100, A127101, A127102, A127104, A127105, A127106, A127107, A127092.

Programs

Mathematica

Select[Range[30000], IntegerQ[(PowerMod[3, #, #^2 ]-1)/#^2 ]&]
Join[{1},Select[Range[335*10^4],PowerMod[3,#,#^2]==1&]] (* Harvey P. Dale, Oct 02 2019 *)

PARI

is(k) = Mod(3, k^2)^k == 1; \\ Amiram Eldar, May 21 2024

Extensions

More terms from Ryan Propper and Alexander Adamchuk, Jan 05 2007