A057957 -id:A057957 - cp's OEIS frontend

A059886 a(n) = |{m : multiplicative order of 4 mod m=n}|.

2, 2, 4, 4, 6, 16, 6, 8, 26, 38, 14, 68, 6, 54, 84, 16, 6, 462, 6, 140, 132, 110, 14, 664, 120, 118, 128, 188, 62, 4456, 6, 96, 364, 118, 498, 7608, 30, 118, 180, 568, 30, 9000, 30, 892, 3974, 494, 62, 5360, 24, 8024, 1524, 892, 62, 9600, 3050, 1784, 372, 446

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) is the number of orders of degree-n monic irreducible polynomials over GF(4).
Also, number of primitive factors of 4^n - 1. - Max Alekseyev, May 03 2022

			a(1) = |{1,3}| = 2, a(2) = |{5,15}| =2, a(3) = |{7,9,21,63}| =4, a(4) = |{17,51,85,255}| = 4.

Max Alekseyev, Table of n, a(n) for n = 1..1128 (first 160 terms from Alois P. Heinz)

Number of primitive factors of b^n - 1: A059499 (b=2), A059885(b=3), this sequence (b=4), A059887 (b=5), A059888 (b=6), A059889 (b=7), A059890 (b=8), A059891 (b=9), A059892 (b=10).
Cf. A000005, A008683, A027377, A053447, A057957, A058948, A112092, A274906.
Column k=4 of A212957.

with(numtheory):
a:= n-> add(mobius(n/d)*tau(4^d-1), d=divisors(n)):
seq(a(n), n=1..60);  # Alois P. Heinz, Oct 12 2012

a[n_] := DivisorSum[n, MoebiusMu[n/#]*DivisorSigma[0, 4^# - 1]&]; Array[a, 100] (* Jean-François Alcover, Nov 11 2015 *)

a(n) = sumdiv(n, d, moebius(n/d) * numdiv(4^d-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = Sum_{ d divides n } mu(n/d)*tau(4^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).

A069061 Sum of divisors of 2^n+1.

Original entry on oeis.org

4, 6, 13, 18, 48, 84, 176, 258, 800, 1302, 2736, 4356, 10928, 20520, 51792, 65538, 174768, 351120, 699056, 1110276, 3100240, 5048232, 11184816, 17041416, 49012992, 82623888, 211053040, 284225796, 727960800, 1494039792, 2863311536, 4301668356, 12611914848, 20788904016

Offset: 1

Benoit Cloitre, Apr 04 2002

nonn

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

Cf. A000051, A000203, A002185, A002587, A002589, A046798, A046799, A053285, A054992, A057957, A059886, A085029, A086257, A112092, A250291, A274906, A295501.

Row sums of A374237.

DivisorSigma[1, 2^Range[50] + 1] (* Paolo Xausa, Jul 05 2024 *)

a(n) = sigma(2^n+1); \\ Michel Marcus, Nov 24 2013

a(n) = sigma(2^n+1).

a(n) = A000203(A000051(n)). - Michel Marcus, Nov 24 2013

More terms from Amiram Eldar, Oct 04 2019

A085029 Number of prime factors of cyclotomic(n,4), which is A019322(n), the value of the n-th cyclotomic polynomial evaluated at x=4.

Original entry on oeis.org

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

Offset: 1

T. D. Noe, Jun 19 2003

nonn

The Mobius transform of this sequence yields A057957, number of prime factors of 4^n-1.

See references at A085021

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

omega(Phi(n,x)): A085021 (x=2), A085028 (x=3), this sequence (x=4), A085030 (x=5), A085031 (x=6), A085032 (x=7), A085033 (x=8), A085034 (x=9), A085035 (x=10).

Cf. A019322, A057957.

Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 4]]][[2]], {n, 1, 100}]

A366602 Number of divisors of 4^n-1.

Original entry on oeis.org

2, 4, 6, 8, 8, 24, 8, 16, 32, 48, 16, 96, 8, 64, 96, 32, 8, 512, 8, 192, 144, 128, 16, 768, 128, 128, 160, 256, 64, 4608, 8, 128, 384, 128, 512, 8192, 32, 128, 192, 768, 32, 9216, 32, 1024, 4096, 512, 64, 6144, 32, 8192, 1536, 1024, 64, 10240, 3072, 2048, 384

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

			a(4)=8 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

Cf. A024036, A000005, A057957, A295501, A366603, A366604.

Cf. A046798, A046801, A366575, A366612, A366621, A366633, A366652, A366661, A070528, A366683, A366709.

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

DivisorSigma[0,4^Range[100]-1] (* Paolo Xausa, Oct 14 2023 *)

PARI
```
a(n) = numdiv(4^n-1);
```

a(n) = sigma0(4^n-1) = A000005(A024036(n)).

a(n) = A046801(2*n) = A046798(n) * A046801(n). - 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

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

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

A059886 a(n) = |{m : multiplicative order of 4 mod m=n}|.

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A069061 Sum of divisors of 2^n+1.

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A085029 Number of prime factors of cyclotomic(n,4), which is A019322(n), the value of the n-th cyclotomic polynomial evaluated at x=4.

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A366602 Number of divisors of 4^n-1.

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

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

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

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