A366620 -id:A366620 - cp's OEIS frontend

A366623 a(n) = phi(6^n-1), where phi is Euler's totient function (A000010).

4, 24, 168, 864, 6200, 30240, 223944, 1119744, 7457184, 37200000, 277618528, 1254113280, 10445497920, 51618196224, 365601600000, 1770091315200, 13439285266176, 62336092492800, 484935499902880, 2179146240000000, 17141125020596640, 86330728271779200

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

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

phi(k^n-1): A053287 (k=2), A295500 (k=3), A295501 (k=4), A295502 (k=5), this sequence (k=6), A366635 (k=7), A366654 (k=8), A366663 (k=9), A295503 (k=10), A366685 (k=11), A366711 (k=12).

Cf. A024062, A000010, A295502, A366620, A366621, A366622.

EulerPhi[6^Range[22] - 1] (* Paul F. Marrero Romero, Oct 23 2023 *)

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

a(n) = A000010(A024062(n)). - Paul F. Marrero Romero, Oct 23 2023

A366621 Number of divisors of 6^n-1.

Original entry on oeis.org

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)).

A366632 Number of distinct prime divisors of 7^n - 1.

Original entry on oeis.org

2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 4, 7, 3, 6, 6, 6, 4, 7, 4, 8, 6, 6, 5, 11, 5, 5, 9, 8, 5, 10, 5, 8, 8, 5, 7, 11, 5, 6, 7, 11, 5, 11, 4, 10, 10, 6, 4, 14, 8, 8, 9, 8, 5, 12, 6, 13, 8, 6, 6, 17, 6, 8, 9, 11, 9, 13, 6, 9, 9, 15, 4, 18, 7, 7, 10, 8, 9, 13, 4, 16, 13

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..388

Cf. A024075, A001221, A057954, A059889, A074249, A218358, A366620, A366633, A366634, A366635.

Cf. A046800, A133801, A366604, A366611, A366620, A366651, A366660, A102347, A366681, A366707.

for(n = 1, 100, print1(omega(7^n - 1), ", "))

a(n) = omega(7^n-1) = A001221(A024075(n)).

A366660 Number of distinct prime divisors of 9^n - 1.

Original entry on oeis.org

1, 2, 3, 3, 3, 5, 3, 5, 6, 5, 5, 7, 3, 6, 8, 6, 6, 9, 5, 7, 8, 8, 4, 12, 7, 6, 11, 9, 7, 12, 6, 7, 10, 9, 8, 12, 6, 8, 12, 11, 6, 14, 4, 12, 16, 7, 8, 15, 10, 12, 13, 9, 6, 15, 11, 14, 13, 10, 5, 18, 5, 10, 16, 8, 9, 15, 6, 13, 13, 15, 7, 19, 7, 10, 19, 13, 11

Offset: 1

Sean A. Irvine, Oct 15 2023

nonn

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

Cf. A024101, A001221, A057952, A366661, A366662, A366663.

Cf. A046800, A133801, A366604, A366611, A366620, A366632, A366651, A102347, A366681, A366707.

for(n = 1, 100, print1(omega(9^n - 1), ", "))

a(n) = omega(9^n-1) = A001221(A024101(n)).

a(n) = A133801(2*n) = A133801(n) + A366580(n) - 1. - Max Alekseyev, Jan 07 2024

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

A366604 Number of distinct prime divisors of 4^n - 1.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 6, 3, 6, 6, 5, 3, 8, 3, 7, 6, 7, 4, 9, 7, 7, 6, 8, 6, 11, 3, 7, 8, 7, 9, 12, 5, 7, 7, 9, 5, 12, 5, 10, 11, 9, 6, 12, 5, 12, 10, 10, 6, 12, 11, 11, 8, 9, 6, 15, 3, 8, 11, 9, 9, 14, 5, 10, 8, 15, 6, 17, 6, 10, 13, 11, 10, 16, 5

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..1128

Cf. A024036, A001221, A057957, A133801, A366602, A366603.

Cf. A046800, A133801, A366611, A366620, A366632, A366651, A366660, A102347, A366681, A366707.

PrimeNu[4^Range[100]-1] (* Paolo Xausa, Oct 14 2023 *)

for(n = 1, 100, print1(omega(4^n - 1), ", "))

from sympy import primenu
def A366604(n): return primenu((1<<(n<<1))-1) # Chai Wah Wu, Oct 15 2023

a(n) = omega(4^n-1) = A001221(A024036(n)).

a(n) = A046800(2*n) = A046799(n) + A046800(n). - Max Alekseyev, Jan 07 2024

A366651 Number of distinct prime divisors of 8^n - 1.

Original entry on oeis.org

1, 2, 2, 4, 3, 4, 3, 6, 3, 6, 4, 8, 4, 6, 6, 9, 5, 6, 4, 11, 6, 8, 4, 12, 7, 7, 6, 12, 6, 11, 3, 12, 8, 10, 10, 12, 6, 8, 9, 15, 5, 11, 5, 14, 10, 8, 6, 17, 5, 13, 8, 16, 8, 12, 10, 17, 7, 10, 6, 21, 5, 7, 9, 15, 8, 15, 6, 19, 9, 20, 7, 18, 7, 12, 14, 16, 9

Offset: 1

Sean A. Irvine, Oct 15 2023

nonn

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

Cf. A024088, A001221, A057953, A366652, A366653, A366654.

Cf. A046800, A133801, A366604, A366611, A366620, A366632, A366660, A102347, A366681, A366707.

for(n = 1, 100, print1(omega(8^n - 1), ", "))

a(n) = omega(8^n-1) = A001221(A024088(n)).

a(n) = A046800(3*n). - Max Alekseyev, Jan 09 2024

A366611 Number of distinct prime divisors of 5^n - 1.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 2, 4, 4, 5, 2, 6, 2, 5, 6, 6, 3, 7, 4, 8, 5, 6, 3, 8, 7, 5, 8, 7, 4, 11, 3, 8, 5, 6, 8, 11, 4, 8, 5, 11, 3, 12, 3, 9, 11, 6, 2, 11, 3, 11, 7, 8, 4, 14, 8, 9, 6, 7, 3, 17, 4, 7, 10, 11, 7, 12, 6, 11, 8, 14, 3, 16, 4, 8, 15, 11, 6, 11, 4, 15

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..502

Cf. A024049, A001221, A057956, A059887, A074479, A143665, A218357, A295502, A366612, A366613.

Cf. A046800, A133801, A366604, A366620, A366632, A366651, A366660, A102347, A366681, A366707.

for(n = 1, 100, print1(omega(5^n - 1), ", "))

a(n) = omega(5^n-1) = A001221(A024049(n)).

A366627 Number of distinct prime divisors of 6^n + 1.

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Oct 14 2023

nonn

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

Cf. A062394, A001221, A057938, A366620, A366628, A366629, A366630.

Cf. A046799, A366580, A366605, A366615, A366636, A366655, A366664, A119704, A366686, A366712.

PrimeNu[6^Range[0,84] + 1] (* Paul F. Marrero Romero, Nov 11 2023 *)

for(n = 0, 100, print1(omega(6^n + 1), ", "))

a(n) = omega(6^n+1) = A001221(A062394(n)).

cp's OEIS Frontend

A366623 a(n) = phi(6^n-1), where phi is Euler's totient function (A000010).

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A366621 Number of divisors of 6^n-1.

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Maple

Mathematica

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

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Maple

Mathematica

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A366632 Number of distinct prime divisors of 7^n - 1.

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PARI

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A366660 Number of distinct prime divisors of 9^n - 1.

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PARI

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A274907 Largest prime factor of 6^n - 1.

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Magma

Mathematica

PARI

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A366604 Number of distinct prime divisors of 4^n - 1.

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Mathematica

PARI

Python

Formula

A366651 Number of distinct prime divisors of 8^n - 1.

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