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

1, 2, 13, 26, 109, 218, 433, 757, 866, 1417, 1514, 2834, 3889, 5629, 7778, 8209, 9841, 11258, 16418, 17497, 19682, 34994, 47197, 50557, 52489, 58321, 70957, 82513, 94394, 101114, 104978, 106717, 116642, 141914, 165026, 213434, 227461, 327781, 423901, 454922, 613561, 655562, 682357, 758173, 847802, 894781, 922441

Offset: 1

Author

Jianing Song, Nov 11 2023

Keywords

• nonn

Comments

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.

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).

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.

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.

Examples

			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.
		

Crossrefs

Cf. A094358 (squarefree divisors of 2^2^i - 1), A357266 (divisors of 3^3^i + 1), A014127.

The subsequence of primes is given by A367648.

Programs

PARI

isA357265(k) = (k%3!=0) && isprimepower(3*znorder(Mod(3,k)))