A057956 -id:A057956 - cp's OEIS frontend

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

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)).

A085030 Number of prime factors of cyclotomic(n,5), which is A019323(n), the value of the n-th cyclotomic polynomial evaluated at x=5.

Original entry on oeis.org

2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 1, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 1, 4, 2, 4, 1, 3, 2, 2, 3, 2, 2, 4, 2, 3, 3, 2, 2, 2, 4, 2, 2, 3, 2, 1, 1, 1, 2, 3, 3, 3, 4, 5, 1, 1, 2, 2, 2, 3, 3, 3, 4, 3, 2, 5, 4, 4, 2, 2, 3, 3, 3, 5, 2, 3, 2, 3, 2, 2, 4, 3, 2, 3, 4, 3, 1, 6, 1, 2, 1, 4, 3, 4, 2, 3, 3, 4, 3

Offset: 1

T. D. Noe, Jun 19 2003

nonn

The Mobius transform of this sequence yields A057956, number of prime factors of 5^n-1.

See references at A085021

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

omega(Phi(n,x)): A085021 (x=2), A085028 (x=3), A085029 (x=4), this sequence (x=5), A085031 (x=6), A085032 (x=7), A085033 (x=8), A085034 (x=9), A085035 (x=10).

Cf. A019323, A057956, A059887, A074479.

Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 5]]][[2]], {n, 1, 100}]

A366612 Number of divisors of 5^n-1.

Original entry on oeis.org

3, 8, 6, 20, 12, 48, 6, 48, 24, 64, 6, 240, 6, 64, 96, 224, 12, 512, 24, 640, 48, 128, 12, 1152, 192, 64, 384, 320, 24, 6144, 12, 1024, 48, 128, 384, 10240, 24, 512, 48, 6144, 12, 18432, 12, 1280, 3072, 128, 6, 10752, 12, 4096, 192, 960, 24, 81920, 576, 1536

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

			a(3)=6 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, A000005, A057956, A059887, A074479, A143665, A295502, A218357, A366611, A366613.

Cf. A046801, A366575, A366602, A366621, A366633, A366652, A366661, A070528, A366683, A366709.

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

Mathematica
```
DivisorSigma[0, 5^Range[100]-1]
```
PARI
```
a(n) = numdiv(5^n-1);
```

a(n) = sigma0(5^n-1) = A000005(A024049(n)).

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)).

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

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

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A085030 Number of prime factors of cyclotomic(n,5), which is A019323(n), the value of the n-th cyclotomic polynomial evaluated at x=5.

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A366612 Number of divisors of 5^n-1.

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Examples

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Programs

Maple

Mathematica

PARI

Formula

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

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PARI

Formula

A366708 Number of prime factors of 12^n - 1 (counted with multiplicity).

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PARI

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

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula