A366622 -id:A366622 - 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)).

A366653 Sum of the divisors of 8^n-1.

Original entry on oeis.org

8, 104, 592, 8736, 38912, 473600, 2466048, 38054016, 155493536, 2015330304, 10359014400, 166290432000, 636328345600, 7645340651520, 42424026529792, 648494317126656, 2599936977797120, 32817383473149440, 164708609085669376, 3010983668199456768

Offset: 1

Sean A. Irvine, Oct 15 2023

nonn

			a(5)=38912 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, A000203, A057953, A059890, A366651, A366652, A366654.

Cf. A075708, A366576, A366603, A366613, A366622, A366634, A366662, A102146, A366684, A366710.

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

Mathematica
```
DivisorSigma[1, 8^Range[30]-1]
```

[sigma(8**n-1, 1) for n in range(1, 21)] # Stefano Spezia, Aug 02 2025

a(n) = sigma(8^n-1) = A000203(A024088(n)).

a(n) = A075708(3*n). - Max Alekseyev, Jan 09 2024

A366662 Sum of the divisors of 9^n-1.

Original entry on oeis.org

15, 186, 1680, 15876, 123690, 1541568, 8992680, 111757968, 967814400, 9366647892, 62424587520, 852903426816, 4766016364260, 55176998178240, 550081165885440, 4829754617483040, 31725040326819840, 471309320999516160, 2535353780263288800, 33995669076586206864

Offset: 1

Sean A. Irvine, Oct 15 2023

nonn

			a(2)=186 because 9^2-1 has divisors {1, 2, 4, 5, 8, 10, 16, 20, 40, 80}.

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

Cf. A024101, A000203, A366660, A366661, A366663.

Cf. A075708, A366576, A366603, A366613, A366622, A366634, A366653, A102146, A366684, A366710.

a:=n->numtheory[sigma](9^n-1):
seq(a(n), n=1..30);

Mathematica
```
DivisorSigma[1, 9^Range[30]-1]
```

a(n) = sigma(9^n-1) = A000203(A024101(n)).

a(n) = A366576(2*n) = A366576(n) * A366578(n) * (2^(4 + A007814(n)) - 1) / (2^(3 + A007814(n)) - 1) / 3. - Max Alekseyev, Jan 07 2024

A366603 Sum of the divisors of 4^n-1.

Original entry on oeis.org

4, 24, 104, 432, 1536, 8736, 22528, 111456, 473600, 1999872, 5909760, 38054016, 89522176, 462274560, 2015330304, 7304603328, 22907191296, 166290432000, 366506672128, 2220409884672, 7645340651520, 29833839544320, 95821839806976, 648494317126656

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

			a(4)=432 because 4^4-1 has divisors {1, 3, 5, 15, 17, 51, 85, 255}.

Max Alekseyev, Table of n, a(n) for n = 1..1128 (terms 1..1062 from Robert Israel)

Cf. A024036, A000203, A057957, A069061, A075708, A295501, A366602, A366604.

Cf. A075708, A366576, A366613, A366622, A366634, A366653, A366662, A102146, A366684, A366710.

a:=n->numtheory[sigma](4^n-1):
seq(a(n), n=1..100);

DivisorSigma[1,4^Range[30]-1] (* Paolo Xausa, Oct 14 2023 *)

a(n) = sigma(4^n-1) = A000203(A024036(n)).

a(n) = A069061(n) * A075708(n). - Robert Israel, Nov 22 2023

A366613 Sum of the divisors of 5^n-1.

Original entry on oeis.org

7, 60, 224, 1736, 6048, 49920, 136724, 1107792, 3718400, 27060480, 85449224, 869499904, 2136230474, 15820920000, 61359427584, 461863805760, 1338408456700, 13177159680000, 33558717136896, 301282248701952, 863701914880000, 6313641012910080, 20863951122979048

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

			a(3)=224 because 5^3-1 has divisors {1, 2, 4, 31, 62, 124}.

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

Cf. A024049, A000203, A057956, A059887, A074479, A143665, A218357, A295502, A366611, A366612.

Cf. A075708, A366576, A366603, A366622, A366634, A366653, A366662, A102146, A366684, A366710.

a:=n->numtheory[sigma](5^n-1):
seq(a(n), n=1..100);

Mathematica
```
DivisorSigma[1, 5^Range[30]-1]
```

a(n) = sigma(5^n-1) = A000203(A024049(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)).

A366634 Sum of the divisors of 7^n-1.

Original entry on oeis.org

12, 124, 780, 7812, 33624, 354640, 1704240, 18929096, 97036800, 800520192, 3958188480, 56928231360, 193778020824, 1830926384640, 11181115146240, 115997032277280, 465294239722800, 5175558387507200, 22852200371636160, 287850454432579584, 1318081737957660000

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

			a(5)=33624 because 7^5-1 has divisors {1, 2, 3, 6, 2801, 5602, 8403, 16806}.

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

Cf. A024075, A000203, A057954, A059887, A074249, A218358, A366632, A366633, A366635.

Cf. A075708, A366576, A366603, A366613, A366622, A366653, A366662, A102146, A366684, A366710.

a:=n->numtheory[sigma](7^n-1):
seq(a(n), n=1..100);

Mathematica
```
DivisorSigma[1, 7^Range[30]-1]
```

a(n) = sigma(7^n-1) = A000203(A024075(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

A366629 Sum of the divisors of 6^n+1.

Original entry on oeis.org

3, 8, 38, 256, 1298, 9792, 52136, 338580, 1778436, 11889152, 62367272, 414625216, 2178461956, 15224775552, 80673299432, 611106029568, 2830769440776, 19344856702976, 115255634181184, 696800841097536, 3748220725527432, 27388329197137920, 135183433256806480

Offset: 0

Sean A. Irvine, Oct 14 2023

nonn

			a(3)=256 because 6^3+1 has divisors {1, 7, 31, 217}.

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

Cf. A062394, A000203, A057938, A366622, A366617, A366627, A366628, A366630, A366670.

Cf. A069061, A366578, A366607, A366638, A366657, A366666, A366668, A366689, A366715.

a:=n->numtheory[sigma](6^n+1):
seq(a(n), n=0..100);

DivisorSigma[1, 6^Range[0, 30] + 1] (* Paolo Xausa, Jul 03 2024 *)

a(n) = sigma(6^n+1) = A000203(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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PARI

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A366621 Number of divisors of 6^n-1.

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Maple

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PARI

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A366653 Sum of the divisors of 8^n-1.

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Maple

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SageMath

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A366662 Sum of the divisors of 9^n-1.

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Maple

Mathematica

Formula

A366603 Sum of the divisors of 4^n-1.

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Maple

Mathematica

Formula

A366613 Sum of the divisors of 5^n-1.

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Maple

Mathematica

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

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PARI

Formula

A366634 Sum of the divisors of 7^n-1.

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