A366720 -id:A366720 - cp's OEIS frontend

A366714 Number of divisors of 12^n+1.

2, 2, 4, 8, 4, 4, 8, 8, 8, 32, 12, 4, 16, 24, 16, 128, 4, 8, 32, 16, 64, 384, 64, 16, 64, 64, 32, 1024, 8, 8, 48, 8, 4, 512, 16, 32, 128, 16, 32, 1536, 16, 32, 64, 32, 16, 4096, 8, 32, 32, 32, 512, 512, 32, 32, 1024, 128, 512, 1536, 192, 64, 1024, 32, 64

Offset: 0

Sean A. Irvine, Oct 17 2023

nonn

			a(4)=4 because 12^4+1 has divisors {1, 89, 233, 20737}.

Max Alekseyev, Table of n, a(n) for n = 0..306

Cf. A178248, A000005, A046798, A344897, A366709, A366712, A366713, A366715, A366716, A366719, A366720, A366688.

a:=n->numtheory[tau](12^n+1):
seq(a(n), n=0..100);

DivisorSigma[0, 12^Range[0, 70] + 1] (* Paolo Xausa, Apr 20 2025 *)

PARI
```
a(n) = numdiv(12^n+1);
```

a(n) = sigma0(12^n+1) = A000005(A178248(n)).

A366716 a(n) = phi(12^n+1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 12, 112, 1296, 20416, 229680, 2306304, 32916240, 400515072, 3863116800, 47825825600, 685853880624, 8732596764672, 97509650382144, 990242755633152, 11148606564480000, 184883057981234176, 2047145911595946000, 20281543142263603200, 294779525244632305920

Offset: 0

Sean A. Irvine, Oct 17 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..306

Cf. A178248, A000010, A366711, A366712, A366713, A366714, A366715, A366719, A366720, A366690.

EulerPhi[12^Range[0,19] + 1] (* Paul F. Marrero Romero, Oct 27 2023 *)

PARI
```
{a(n) = eulerphi(12^n+1)}
```

a(n) = A000010(A178248(n)). - Paul F. Marrero Romero, Oct 27 2023

A366715 Sum of the divisors of 12^n+1.

Original entry on oeis.org

3, 14, 180, 2240, 21060, 267988, 3706920, 38773952, 459970056, 6692483840, 79425033660, 800162860756, 9101898907920, 117326869641600, 1596198064568400, 20655000929239040, 184885459808838660, 2390210102271311936, 33504016991491136160, 344201347103878781440

Offset: 0

Sean A. Irvine, Oct 17 2023

nonn

			a(4)=21060 because 12^4+1 has divisors {1, 89, 233, 20737}.

Max Alekseyev, Table of n, a(n) for n = 0..306

Cf. A178248, A000203, A366710, A366712, A366713, A366714, A366716, A366719, A366720, A366689.

a:=n->numtheory[sigma](12^n+1):
seq(a(n), n=0..100);

a(n) = sigma(12^n+1) = A000203(A178248(n)).

A366719 Smallest prime dividing 12^n + 1.

Original entry on oeis.org

2, 13, 5, 7, 89, 13, 5, 13, 17, 7, 5, 13, 89, 13, 5, 7, 153953, 13, 5, 13, 41, 7, 5, 13, 17, 13, 5, 7, 89, 13, 5, 13, 769, 7, 5, 13, 89, 13, 5, 7, 17, 13, 5, 13, 89, 7, 5, 13, 7489, 13, 5, 7, 89, 13, 5, 13, 17, 7, 5, 13, 41, 13, 5, 7, 36097, 13, 5, 13, 89, 7

Offset: 0

Sean A. Irvine, Oct 17 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..511

Cf. A002586, A038371, A178248, A366712, A366713, A366714, A366715, A366716, A366717, A366720.

Table[FactorInteger[12^n + 1][[1,1]], {n, 0, 69}] (* Paul F. Marrero Romero, Oct 25 2023 *)

a(n) = A020639(A178248(n)). - Paul F. Marrero Romero, Oct 25 2023

A366713 Number of prime factors of 12^n + 1 (counted with multiplicity).

Original entry on oeis.org

1, 1, 2, 3, 2, 2, 3, 3, 3, 5, 4, 2, 4, 5, 4, 7, 2, 3, 5, 4, 6, 9, 6, 4, 6, 6, 5, 10, 3, 3, 6, 3, 2, 9, 4, 5, 7, 4, 5, 11, 4, 5, 6, 5, 4, 12, 3, 5, 5, 5, 10, 9, 5, 5, 10, 7, 9, 11, 8, 6, 10, 5, 6, 15, 5, 9, 11, 4, 5, 12, 10, 3, 10, 5, 8, 17, 5, 6, 9, 4, 6, 15

Offset: 0

Sean A. Irvine, Oct 17 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..306

Cf. A178248, A001222, A054992, A057934, A057935, A057936, A057937, A057938, A057939, A057940, A057941, A366712, A366714, A366715, A366716, A366719, A366720, A366708, A366687.

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

a(n) = bigomega(12^n+1) = A001222(A178248(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

A366714 Number of divisors of 12^n+1.

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A366716 a(n) = phi(12^n+1), where phi is Euler's totient function (A000010).

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A366715 Sum of the divisors of 12^n+1.

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A366719 Smallest prime dividing 12^n + 1.

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A366713 Number of prime factors of 12^n + 1 (counted with multiplicity).

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A366718 Largest prime factor of 12^n - 1.

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