A057955 -id:A057955 - cp's OEIS frontend

A366621 Number of divisors of 6^n-1.

2, 4, 4, 8, 6, 16, 4, 16, 16, 48, 8, 128, 8, 48, 48, 64, 32, 128, 8, 384, 16, 32, 32, 512, 32, 128, 64, 384, 4, 1536, 8, 512, 64, 256, 96, 8192, 64, 64, 64, 3072, 8, 768, 32, 512, 1536, 256, 16, 8192, 32, 512, 512, 2048, 16, 2048, 96, 12288, 128, 64, 16

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

			a(4)=8 because 6^4-1 has divisors {1, 5, 7, 35, 37, 185, 259, 1295}.

Max Alekseyev, Table of n, a(n) for n = 1..430

Cf. A024062, A000005, A057955, A059888, A274907, A366613, A366620, A366622, A366623.

Cf. A046801, A366575, A366602, A366612, A366633, A366652, A366661, A070528, A366683, A366709.

a:=n->numtheory[tau](6^n-1):
seq(a(n), n=1..100);

Mathematica
```
DivisorSigma[0, 6^Range[100]-1]
```
PARI
```
a(n) = numdiv(6^n-1);
```

a(n) = sigma0(6^n-1) = A000005(A024062(n)).

A366622 Sum of the divisors of 6^n-1.

Original entry on oeis.org

6, 48, 264, 1824, 9672, 67584, 335928, 2367552, 13031040, 94708224, 454285152, 3523559424, 15677418768, 113738502240, 599516366592, 4210539708672, 20465720064000, 154928015278080, 735060126170880, 5906693566844928, 26937015875831424, 188358079273592832

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

			a(4)=1824 because 6^4-1 has divisors {1, 5, 7, 35, 37, 185, 259, 1295}.

Max Alekseyev, Table of n, a(n) for n = 1..430

Cf. A024062, A000203, A057955, A059888, A274907, A366620, A366621, A366623, A366629.

Cf. A075708, A366576, A366603, A366613, A366634, A366653, A366662, A102146, A366684, A366710.

a:=n->numtheory[sigma](6^n-1):
seq(a(n), n=1..100);

Mathematica
```
DivisorSigma[1, 6^Range[30]-1]
```

a(n) = sigma(6^n-1) = A000203(A024062(n)).

A085031 Number of prime factors of cyclotomic(n,6), which is A019324(n), the value of the n-th cyclotomic polynomial evaluated at x=6.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 2, 3, 2, 2, 4, 1, 2, 2, 1, 1, 4, 1, 3, 3, 2, 2, 1, 1, 2, 3, 2, 2, 3, 3, 5, 2, 2, 2, 2, 1, 4, 3, 3, 2, 3, 2, 3, 1, 3, 3, 3, 2, 2, 4, 3, 3, 3, 4, 3, 1, 4, 3, 4, 3, 2, 2, 2, 5, 1, 3, 4, 3, 3, 2, 2, 4, 3, 3, 2, 3, 7, 2, 3, 1, 4, 2, 3, 1, 2

Offset: 1

T. D. Noe, Jun 19 2003

nonn

The Mobius transform of this sequence yields A057955, number of prime factors of 6^n-1.

See references at A085021

Max Alekseyev, Table of n, a(n) for n = 1..430

omega(Phi(n,x)): A085021 (x=2), A085028 (x=3), A085029 (x=4), A085030 (x=5), this sequence (x=6), A085032 (x=7), A085033 (x=8), A085034 (x=9), A085035 (x=10).

Cf. A019324, A057955, A059888, A274907.

Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 6]]][[2]], {n, 1, 100}]

A366620 Number of distinct prime divisors of 6^n - 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 4, 5, 3, 7, 3, 5, 5, 6, 5, 7, 3, 8, 4, 5, 5, 9, 4, 7, 6, 8, 2, 10, 3, 9, 6, 8, 6, 13, 6, 6, 6, 11, 3, 9, 5, 9, 10, 8, 4, 13, 5, 8, 9, 11, 4, 11, 6, 13, 7, 6, 4, 19, 4, 5, 10, 12, 8, 12, 3, 11, 8, 16, 2, 18, 5, 10, 10, 9, 6, 15, 4, 16, 8

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..430

Cf. A024062, A001221, A057955, A059888, A274907, A366621, A366622, A366623.

Cf. A046800, A133801, A366604, A366611, A366632, A366651, A366660, A102347, A366681, A366707.

for(n = 1, 100, print1(omega(6^n - 1), ", "))

a(n) = omega(6^n-1) = A001221(A024062(n)).

A274907 Largest prime factor of 6^n - 1.

Original entry on oeis.org

5, 7, 43, 37, 311, 43, 55987, 1297, 2467, 311, 3154757, 97, 760891, 55987, 1201, 98801, 30839, 46441, 638073026189, 6781, 1822428931, 51828151, 7505944891, 1678321, 40185601, 760891, 623067280651, 5030761, 7369130657357778596659, 1950271, 49744740983476472807

Offset: 1

Vincenzo Librandi, Jul 11 2016

nonn

			6^5 - 1 = 7775 = 5*5*311, so a(5) = 311.

Max Alekseyev, Table of n, a(n) for n = 1..430 (terms 1..100 from Vincenzo Librandi, term 101..420 from Amiram Eldar)
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

Cf. A006530, A057955, A059888, A085031, A024062, A366620, A366621, A366622, A366623.

Cf. similar sequences listed in A274906.

[Maximum(PrimeDivisors(6^n-1)): n in [1..40]];

Table[FactorInteger[6^n - 1][[-1, 1]], {n, 40}]

a(n) = vecmax(factor(6^n-1)[,1]); \\ Michel Marcus, Jul 13 2016

a(n) = A006530(A024062(n)). - Michel Marcus, Jul 11 2016

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)).

A366682 Number of prime factors of 11^n - 1 (counted with multiplicity).

Original entry on oeis.org

2, 5, 4, 7, 4, 9, 4, 9, 5, 8, 4, 13, 4, 8, 7, 12, 3, 12, 3, 11, 10, 11, 5, 17, 8, 10, 6, 13, 4, 15, 5, 15, 9, 9, 8, 17, 6, 10, 12, 15, 9, 17, 4, 15, 9, 12, 5, 24, 7, 14, 9, 13, 6, 16, 10, 19, 8, 10, 5, 21, 5, 12, 16, 19, 8, 22, 6, 15, 10, 19, 7, 24, 3, 11, 15

Offset: 1

Sean A. Irvine, Oct 16 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..316

Cf. A024127, A001222, A366681, A366683, A366684, A366685, A366687.

Cf. A046051, A057958, A057957, A057956, A057955, A057954, A057953, A057952, A057951, A366708.

Mathematica
```
PrimeOmega[11^Range[70]-1]
```
PARI
```
a(n)=bigomega(11^n-1)
```

a(n) = bigomega(11^n-1) = A001222(A024127(n)).

cp's OEIS Frontend

A366621 Number of divisors of 6^n-1.

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Maple

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A366622 Sum of the divisors of 6^n-1.

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Maple

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A085031 Number of prime factors of cyclotomic(n,6), which is A019324(n), the value of the n-th cyclotomic polynomial evaluated at x=6.

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Mathematica

A366620 Number of distinct prime divisors of 6^n - 1.

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PARI

Formula

A274907 Largest prime factor of 6^n - 1.

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Magma

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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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Formula

A366682 Number of prime factors of 11^n - 1 (counted with multiplicity).

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula