A367648 -id:A367648 - cp's OEIS frontend

A367649 Primes p such that the multiplicative order of 3 modulo p is 2 times a power of 3.

7, 19, 37, 163, 487, 1297, 1459, 2917, 19441, 19927, 39367, 59779, 131221, 208657, 224209, 572023, 2051893, 5062663, 8503057, 19131877, 44457337, 86093443, 113863969, 133923133, 258280327, 565571323, 600830137, 859270843, 1319934691, 4161183031, 5366491219, 5879415781

Offset: 1

Jianing Song, Nov 25 2023

nonn

Odd prime factors of numbers of the form 3^3^i + 1: for odd primes p, p divides 3^3^i + 1 if and only if the multiplicative order of 3 modulo p is 2 times a power of 3 not exceeding 3^i.

			37 is a term since the multiplicative order of 3 modulo 37 is 18 = 2*3^2, which means that 37 is a factor of 3^3^2 + 1.
163 is a term since the multiplicative order of 3 modulo 163 is 162 = 2*3^4, which means that 163 is a factor of 3^3^4 + 1.

Subsequence of A367266.

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)), A367648 (ord(3,p) being a power of 3, prime factors of numbers of the form 3^3^i - 1).

isA367649(n) = my(d); isprime(n) && (n!=3) && ((d=znorder(Mod(3,n)))%2==0) && isprimepower(3*d/2)

a(28)-a(31) from Chai Wah Wu, Nov 26 2023

a(32) from Jinyuan Wang, Jan 29 2025

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

Original entry on oeis.org

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

Jianing Song, Nov 11 2023

nonn

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.

			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.

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.

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

cp's OEIS Frontend

A367649 Primes p such that the multiplicative order of 3 modulo p is 2 times a power of 3.

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