A366709 -id:A366709 - cp's OEIS frontend

A366652 Number of divisors of 8^n-1.

2, 6, 4, 24, 8, 32, 12, 96, 8, 96, 16, 512, 16, 144, 64, 768, 32, 160, 16, 4608, 96, 384, 16, 8192, 128, 192, 64, 9216, 64, 4096, 8, 6144, 256, 1536, 1536, 10240, 64, 384, 512, 73728, 32, 6144, 32, 24576, 1024, 384, 64, 262144, 64, 12288, 256, 147456, 256

Offset: 1

Sean A. Irvine, Oct 15 2023

nonn

			a(5)=8 because 8^5-1 has divisors {1, 7, 31, 151, 217, 1057, 4681, 32767}.

Max Alekseyev, Table of n, a(n) for n = 1..500

Cf. A024088, A000005, A057953, A059890, A366651, A366653, A366654.

Cf. A046801, A366575, A366602, A366612, A366621, A366633, A366661, A070528, A366683, A366709.

a:=n->numtheory[tau](8^n-1):
seq(a(n), n=1..100);

Mathematica
```
DivisorSigma[0, 8^Range[100]-1]
```
PARI
```
a(n) = numdiv(8^n-1);
```

a(n) = sigma0(8^n-1) = A000005(A024088(n)).

a(n) = A046801(3*n). - Max Alekseyev, Jan 09 2024

A366575 Number of divisors of 3^n - 1.

Original entry on oeis.org

2, 4, 4, 10, 6, 16, 4, 24, 8, 24, 8, 80, 4, 16, 24, 112, 8, 128, 8, 180, 16, 64, 8, 384, 24, 16, 64, 160, 16, 768, 16, 256, 32, 128, 48, 1280, 8, 64, 96, 864, 16, 768, 8, 640, 384, 32, 32, 14336, 128, 384, 64, 160, 16, 4096, 128, 1536, 128, 256, 8, 23040, 8

Offset: 1

Sean A. Irvine, Oct 13 2023

nonn

			a(4)=10 because 3^4-1 has divisors {1, 2, 4, 5, 8, 10, 16, 20, 40, 80}.

Max Alekseyev, Table of n, a(n) for n = 1..690

Cf. A024023, A000005, A046801, A133801, A295500, A366576.

Cf. A046801, A366602, A366612, A366621, A366633, A366652, A366661, A070528, A366683, A366709.

a:=n->numtheory[tau](3^n-1):
seq(a(n), n=1..100);

DivisorSigma[0,3^Range[100]-1] (* Paolo Xausa, Oct 15 2023 *)

a(n) = sigma0(3^n-1) = A000005(A024023).

A366708 Number of prime factors of 12^n - 1 (counted with multiplicity).

Original entry on oeis.org

1, 2, 2, 4, 2, 5, 3, 6, 4, 4, 4, 8, 3, 6, 6, 9, 3, 9, 2, 8, 5, 6, 4, 12, 4, 8, 6, 10, 5, 13, 5, 11, 8, 6, 9, 14, 3, 6, 7, 14, 4, 14, 5, 12, 12, 8, 3, 18, 5, 10, 6, 13, 7, 16, 8, 13, 7, 8, 4, 19, 4, 8, 8, 13, 8, 17, 5, 10, 7, 14, 4, 21, 3, 7, 11, 11, 11, 18, 4

Offset: 1

Sean A. Irvine, Oct 17 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..310

Cf. A024140, A001222, A046051, A057951, A057952, A057953, A057954, A057955, A057956, A057957, A057958, A250288, A252170, A366707, A366709, A366682.

Mathematica
```
PrimeOmega[12^Range[70]-1]
```
PARI
```
a(n)=bigomega(12^n-1)
```

a(n) = bigomega(12^n-1) = A001222(A024140(n)).

A366718 Largest prime factor of 12^n - 1.

Original entry on oeis.org

11, 13, 157, 29, 22621, 157, 4943, 233, 80749, 22621, 266981089, 20593, 20369233, 13063, 22621, 260753, 74876782031, 80749, 29043636306420266077, 85403261, 8177824843189, 57154490053, 321218438243, 2227777, 12629757106815551, 20369233, 86769286104133

Offset: 1

Sean A. Irvine, Oct 17 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..310
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

Cf. A024140, A006530, A005420, A074477, A274906, A074479, A274907, A074249, A274908, A274909, A005422, A274910, A366707, A366708, A366709, A366710, A366711, A366717, A366720.

[Maximum(PrimeDivisors(12^n-1)): n in [1..40]];

Table[FactorInteger[12^n - 1][[-1, 1]], {n, 40}]

a(n) = A006530(A024140(n)).

cp's OEIS Frontend

A366652 Number of divisors of 8^n-1.

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A366575 Number of divisors of 3^n - 1.

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A366708 Number of prime factors of 12^n - 1 (counted with multiplicity).

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Mathematica

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A366718 Largest prime factor of 12^n - 1.

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