A367648 Primes p such that the multiplicative order of 3 modulo p is a power of 3.

2, 13, 109, 433, 757, 3889, 8209, 17497, 52489, 58321, 70957, 1190701, 1705861, 2598157, 6627097, 13463173, 57395629, 23245229341, 79320757897, 1069604540569, 1631815099669, 5774114968057, 8635817966221, 23765922477217, 43781455818469, 307283335691329

Offset: 1

Author

Jianing Song, Nov 25 2023

Keywords

• nonn
• hard

Comments

Prime factors of numbers of the form 3^3^i - 1: p divides 3^3^i - 1 if and only if the multiplicative order of 3 modulo p is a power of 3 not exceeding 3^i.

Examples

			13 is a term since the multiplicative order of 3 modulo 13 is 3 = 3^1, which means that 13 is a factor of 3^3^1 - 1.
109 is a term since the multiplicative order of 3 modulo 109 is 27 = 3^3, which means that 109 is a factor of 3^3^3 - 1.
		

Crossrefs

Subsequence of A367265.

Cf. A023394 (ord(2,p) being a power of 2, prime factors of numbers of the form 2^2^i - 1 (or of the form 2^2^i + 1)), A367649 (ord(3,p) being 2 times a power of 3, prime factors of numbers of the form 3^3^i + 1).

Programs

PARI

isA367648(n) = isprime(n) && (n!=3) && isprimepower(3*znorder(Mod(3,n)))

Extensions

a(18)-a(19) from Michel Marcus, Nov 27 2023

a(20)-a(25) from Max Alekseyev, Jul 22 2024

a(26) from Jinyuan Wang, Jan 29 2025