cp's OEIS Frontend

A367265 Numbers k such that there exists i >= 1 such that k divides 3^3^i - 1.

%I A367265 #7 Nov 25 2023 23:55:10
%S A367265 1,2,13,26,109,218,433,757,866,1417,1514,2834,3889,5629,7778,8209,
%T A367265 9841,11258,16418,17497,19682,34994,47197,50557,52489,58321,70957,
%U A367265 82513,94394,101114,104978,106717,116642,141914,165026,213434,227461,327781,423901,454922,613561,655562,682357,758173,847802,894781,922441
%N A367265 Numbers k such that there exists i >= 1 such that k divides 3^3^i - 1.
%C A367265 Note that 3^3^i - 1 divides 3^3^(i+1) - 1, so this sequence is also numbers k such that k divides 3^3^i - 1 for all sufficiently large i.
%C A367265 Also numbers k such that there exists i >= 1 such that k divides 3^^i - 1, where 3^^i = 3^3^...^3 (i times) = A014220(i-1).
%C A367265 Also numbers k such that ord(3,k) is a power of 3, where ord(a,k) is the multiplicative order of a modulo k: 3^3^i == 1 (mod k) if and only if ord(3,k) divides 3^i, so such i exists if and only if ord(3,k) is a power of 3.
%C A367265 If a term k is not squarefree, then it is divisible by p^2, where p is a Wieferich prime to base 3 (A014127) such that ord(3,p) is a power of 3. No such p is known.
%e A367265 Suppose that q is an odd prime power such that ord(3,q) = 3^e. e = 1 gives q = 13; e = 2 gives q = 757; e = 3 gives q = 109, 433, 8209; e = 4 gives q = 3889, 1190701, 12557612956332313.
%o A367265 (PARI) isA357265(k) = (k%3!=0) && isprimepower(3*znorder(Mod(3,k)))
%Y A367265 Cf. A094358 (squarefree divisors of 2^2^i - 1), A357266 (divisors of 3^3^i + 1), A014127.
%Y A367265 The subsequence of primes is given by A367648.
%K A367265 nonn
%O A367265 1,2
%A A367265 _Jianing Song_, Nov 11 2023