A274908 -id:A274908 - cp's OEIS frontend

A057953 Number of prime factors of 8^n - 1 (counted with multiplicity).

1, 3, 2, 5, 3, 6, 4, 7, 3, 7, 4, 10, 4, 8, 6, 10, 5, 9, 4, 13, 7, 9, 4, 14, 7, 8, 6, 14, 6, 13, 3, 13, 8, 11, 11, 15, 6, 9, 9, 17, 5, 14, 5, 15, 10, 9, 6, 19, 7, 14, 8, 18, 8, 16, 10, 19, 7, 11, 6, 24, 5, 8, 10, 16, 8, 17, 6, 20, 9, 22, 7, 21, 7, 13, 14, 17, 10, 16, 8, 23, 10, 12, 5, 24

Offset: 1

Patrick De Geest, Nov 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..500 (first 402 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n-1): A057951 (b=10), A057952 (b=9), this sequence (b=8), A057954 (b=7), A057955 (b=6), A057956 (b=5), A057957 (b=4), A057958 (b=3), A046051 (b=2).

Cf. A001222, A024088, A085033, A274908.

f:=func; [f(8^n - 1):n in [1..90]]; // Marius A. Burtea, Feb 02 2020

PrimeOmega/@(8^Range[90]-1) (* Harvey P. Dale, May 24 2018 *)

Mobius transform of A085033. - T. D. Noe, Jun 19 2003

a(n) = A001222(A024088(n)) = A046051(3*n). - Amiram Eldar, Feb 02 2020

A274906 Largest prime factor of 4^n - 1.

Original entry on oeis.org

3, 5, 7, 17, 31, 13, 127, 257, 73, 41, 683, 241, 8191, 127, 331, 65537, 131071, 109, 524287, 61681, 5419, 2113, 2796203, 673, 4051, 8191, 262657, 15790321, 3033169, 1321, 2147483647, 6700417, 599479, 131071, 122921, 38737, 616318177, 525313, 22366891

Offset: 1

Vincenzo Librandi, Jul 11 2016

nonn

			4^7 - 1 = 16383 = 3*43*127, so a(7) = 127

Max Alekseyev, Table of n, a(n) for n = 1..1128
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

Second bisection of A005420. - Michel Marcus, Jul 13 2016
Cf. largest prime factor of k^n-1: A005420 (k=2), A074477 (k=3), this sequence (k=4), A074479 (k=5), A274907 (k=6), A074249 (k=7), A274908 (k=8), A274909 (k=9), A005422 (k=10), A274910 (k=11).
Cf. A006530, A024036.

[Maximum(PrimeDivisors(4^n-1)): n in [1..40]];

Table[FactorInteger[4^n - 1][[-1, 1]], {n, 40}]

a(n) = A006530(A024036(n)). - Michel Marcus, Jul 11 2016

a(n) = max(A002587(n),A005420(n)). - Max Alekseyev, Apr 25 2022

Terms to a(100) in b-file from Vincenzo Librandi, Jul 13 2016
a(101)-a(603) in b-file from Amiram Eldar, Feb 08 2020
a(604)-a(1128) in b-file from Max Alekseyev, Jul 25 2023, Mar 15 2025

A059890 a(n) = |{m : multiplicative order of 8 mod m = n}|.

Original entry on oeis.org

2, 4, 2, 18, 6, 24, 10, 72, 4, 84, 14, 462, 14, 128, 54, 672, 30, 124, 14, 4494, 82, 364, 14, 7608, 120, 172, 56, 9054, 62, 3920, 6, 5376, 238, 1500, 1518, 9600, 62, 364, 494, 69048, 30, 5892, 30, 24174, 956, 364, 62, 253280, 52, 12072, 222, 147246, 254, 12072

Offset: 1

Vladeta Jovovic, Feb 06 2001

nonn

The multiplicative order of a mod m, gcd(a,m)=1, is the smallest natural number d for which a^d = 1 (mod m). a(n) = number of orders of degree-n monic irreducible polynomials over GF(8).

Also, number of primitive factors of 8^n - 1. - Max Alekseyev, May 03 2022

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

Number of primitive factors of b^n - 1: A059499 (b=2), A059885(b=3), A059886 (b=4), A059887 (b=5), A059888 (b=6), A059889 (b=7), this sequence (b=8), A059891 (b=9), A059892 (b=10).
Cf. A000005, A008683, A027380, A053451, A057953, A058946, A274908.
Column k=8 of A212957.

with(numtheory):
a:= n-> add(mobius(n/d)*tau(8^d-1), d=divisors(n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Oct 12 2012

a[n_] := Sum[MoebiusMu[n/d]*DivisorSigma[0, 8^d-1], {d, Divisors[n]}];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Jan 13 2025, after Alois P. Heinz *)

a(n) = sumdiv(n, d, moebius(n/d) * numdiv(8^d-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = Sum_{ d divides n } mu(n/d)*tau(8^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).

A366654 a(n) = phi(8^n-1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

6, 36, 432, 1728, 27000, 139968, 1778112, 6635520, 113467392, 534600000, 6963536448, 26121388032, 465193834560, 2427720325632, 28548223200000, 109586090557440, 1910296842179040, 9618417501143040, 123523151337020736, 406467072000000000, 7713001620195508224

Offset: 1

Sean A. Irvine, Oct 15 2023

nonn

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

phi(k^n-1): A053287 (k=2), A295500 (k=3), A295501 (k=4), A295502 (k=5), A366623 (k=6), A366635 (k=7), this sequence (k=8), A366663 (k=9), A295503 (k=10), A366685 (k=11), A366711 (k=12).

Cf. A000010, A010705, A024088, A057953, A059890, A274908, A366651, A366652, A366653.

Mathematica
```
EulerPhi[8^Range[30] - 1]
```
PARI
```
{a(n) = eulerphi(8^n-1)}
```

from sympy import totient
def A366654(n): return totient((1<<3*n)-1) # Chai Wah Wu, Oct 15 2023

a(n) = A053287(3*n). - Max Alekseyev, Jan 09 2024

A379641 Smallest primitive prime factor of 8^n-1.

Original entry on oeis.org

7, 3, 73, 5, 31, 19, 127, 17, 262657, 11, 23, 37, 79, 43, 631, 97, 103, 87211, 32377, 41, 92737, 67, 47, 433, 601, 2731, 2593, 29, 233, 18837001, 2147483647, 193, 199, 307, 71, 246241, 223, 571, 937, 61681, 13367, 77158673929, 431, 397, 271, 139, 2351, 577

Offset: 1

Sean A. Irvine, Dec 28 2024

nonn

Also, smallest prime p such that 1/p has octal period n.

Max Alekseyev, Table of n, a(n) for n = 1..548 (first 500 terms from Sean A. Irvine)

Cf. A112927 (base 2), A143663 (base 3), A112092 (base 4), A143665 (base 5), A379639 (base 6), A379640 (base 7), A379641 (base 8), A379642 (base 9), A007138 (base 10), A379644 (base 11), A252170 (base 12).

Cf. A274908.

listap(nn) = {prf = []; for (n=1, nn, vp = (factor(8^n-1)[, 1])~; f = setminus(Set(vp), Set(prf)); prf = concat(prf, f); print1(vecmin(Vec(f)), ", "); ); }

A085033 Number of prime factors of cyclotomic(n,8), which is A019326(n), the value of the n-th cyclotomic polynomial evaluated at x=8.

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 3, 2, 1, 2, 3, 2, 3, 2, 2, 3, 4, 2, 3, 4, 2, 3, 3, 2, 4, 2, 3, 4, 5, 1, 2, 3, 3, 4, 5, 2, 5, 3, 4, 2, 4, 1, 4, 4, 3, 3, 5, 2, 3, 3, 2, 8, 7, 4, 4, 3, 2, 3, 5, 3, 4, 3, 2, 3, 2, 2, 5, 7, 4, 5, 6, 2, 6, 5, 4, 6, 3, 1, 7, 3, 4, 5, 4, 2

Offset: 1

T. D. Noe, Jun 19 2003

nonn

The Mobius transform of this sequence yields A057953, number of prime factors of 8^n-1.

See references at A085021

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

omega(Phi(n,x)): A085021 (x=2), A085028 (x=3), A085029 (x=4), A085030 (x=5), A085031 (x=6), A085032 (x=7), this sequence (x=8), A085034 (x=9), A085035 (x=10).

Cf. A019326, A057953, A059890, A274908.

Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 8]]][[2]], {n, 1, 100}]

A366718 Largest prime factor of 12^n - 1.

Original entry on oeis.org

11, 13, 157, 29, 22621, 157, 4943, 233, 80749, 22621, 266981089, 20593, 20369233, 13063, 22621, 260753, 74876782031, 80749, 29043636306420266077, 85403261, 8177824843189, 57154490053, 321218438243, 2227777, 12629757106815551, 20369233, 86769286104133

Offset: 1

Sean A. Irvine, Oct 17 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..310
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

Cf. A024140, A006530, A005420, A074477, A274906, A074479, A274907, A074249, A274908, A274909, A005422, A274910, A366707, A366708, A366709, A366710, A366711, A366717, A366720.

[Maximum(PrimeDivisors(12^n-1)): n in [1..40]];

Table[FactorInteger[12^n - 1][[-1, 1]], {n, 40}]

a(n) = A006530(A024140(n)).

cp's OEIS Frontend

A057953 Number of prime factors of 8^n - 1 (counted with multiplicity).

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A274906 Largest prime factor of 4^n - 1.

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A059890 a(n) = |{m : multiplicative order of 8 mod m = n}|.

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A366654 a(n) = phi(8^n-1), where phi is Euler's totient function (A000010).

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A379641 Smallest primitive prime factor of 8^n-1.

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A085033 Number of prime factors of cyclotomic(n,8), which is A019326(n), the value of the n-th cyclotomic polynomial evaluated at x=8.

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Mathematica

A366718 Largest prime factor of 12^n - 1.

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Magma

Mathematica

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