A366623 -id:A366623 - cp's OEIS frontend

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

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)).

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)).

A366630 a(n) = phi(6^n+1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 6, 36, 180, 1296, 6000, 41472, 230496, 1580800, 8359200, 58579200, 310968900, 2175102720, 10971642240, 76065091200, 351048600000, 2811459796992, 14508487949472, 88870766837760, 522016066337712, 3564233663616000, 17479898551382400, 128060205344805888

Offset: 0

Sean A. Irvine, Oct 14 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..430

Cf. A062394, A000010, A366623, A366618, A366627, A366628, A366629, A366670.

Cf. A053285, A366579, A366608, A366639, A366658, A366667, A366669, A366690, A366716.

EulerPhi[6^Range[0, 22] + 1] (* Paul F. Marrero Romero, Oct 17 2023 *)

PARI
```
{a(n) = eulerphi(6^n+1)}
```

a(n) = A000010(A062394(n)). - Paul F. Marrero Romero, Oct 17 2023

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

cp's OEIS Frontend

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

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

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A366630 a(n) = phi(6^n+1), where phi is Euler's totient function (A000010).

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A274907 Largest prime factor of 6^n - 1.

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