A274909 -id:A274909 - cp's OEIS frontend

A057958 Number of prime factors of 3^n - 1 (counted with multiplicity).

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

Offset: 1

Patrick De Geest, Nov 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..690 (first 660 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n-1): A057951 (b=10), A057952 (b=9), A057953 (b=8), A057954 (b=7), A057955 (b=6), A057956 (b=5), A057957 (b=4), this sequence (b=3), A046051 (b=2).

Cf. A001222, A024023, A085028, A002591 A074477, A057952, A133801, A274909.

a(n)=bigomega(3^n-1) \\ Charles R Greathouse IV, Sep 14 2015

Mobius transform of A085028. - T. D. Noe, Jun 19 2003

a(n) = A001222(A024023(n)). - Amiram Eldar, Feb 01 2020

Offset corrected by Amiram Eldar, Feb 01 2020

A057952 Number of prime factors of 9^n - 1 (counted with multiplicity).

Original entry on oeis.org

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

Offset: 1

Patrick De Geest, Nov 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..690 (first 330 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n-1): A046051 (b=2), A057958 (b=3), A057957 (b=4), A057956 (b=5), A057955 (b=6), A057954 (b=7), A057953 (b=8), this sequence (b=9), A057951 (b=10), A366682 (b=11), A366708 (b=12).

Cf. A001222, A002591, A024101, A074477, A085034, A133801, A274909, A366660, A366661, A366662.

PrimeOmega[Table[9^n - 1, {n, 1, 30}]] (* Amiram Eldar, Feb 02 2020 *)

Mobius transform of A085034. - T. D. Noe, Jun 19 2003
a(n) = A001222(A024101(n)) = A057958(2*n). - Amiram Eldar, Feb 02 2020
a(n) = A057941(n) + A057958(n). - Max Alekseyev, Jan 07 2024

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

A366579 a(n) = phi(3^n+1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 2, 4, 12, 40, 120, 288, 1092, 3072, 7776, 23600, 87120, 259200, 797160, 1847104, 5832000, 21523360, 63672480, 152845056, 580921200, 1700870400, 4368821184, 12550120000, 47071589412, 130459631616, 413939700000, 997562438080, 3012122557440, 11159367815680

Offset: 0

Sean A. Irvine, Oct 13 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..691

Cf. A000010, A002592, A034472, A053285, A057935, A057941, A074476, A274909, A295500, A366577, A366578, A366580.

EulerPhi[3^Range[0,30]+1] (* Paolo Xausa, Oct 15 2023 *)

PARI
```
{a(n) = eulerphi(3^n+1)}
```

a(n) = phi(3^n+1) = A000010(A034472(n)).

A366580 Number of distinct prime divisors of 3^n + 1.

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Oct 13 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..691

Cf. A001221, A002592, A034472, A046799, A057935, A057941, A074476, A133801, A274909, A366577, A366578, A366579.

PrimeNu[3^Range[0,100]+1] (* Paolo Xausa, Oct 14 2023 *)

for(n = 0, 100, print1(omega(3^n + 1), ", "))

a(n) = omega(3^n+1) = A001221(A034472(n)).

A379642 Smallest primitive prime factor of 9^n-1.

Original entry on oeis.org

2, 5, 7, 41, 11, 73, 547, 17, 19, 1181, 23, 6481, 398581, 29, 31, 21523361, 103, 530713, 1597, 42521761, 43, 5501, 47, 97, 151, 53, 109, 430697, 59, 47763361, 683, 926510094425921, 25411, 956353, 71, 282429005041, 18427, 5301533, 79, 14401, 83, 2857, 431, 89

Offset: 1

Sean A. Irvine, Dec 28 2024

nonn

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

Max Alekseyev, Table of n, a(n) for n = 1..730 (first 690 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. A274909.

listap(nn) = {prf = []; for (n=1, nn, vp = (factor(9^n-1)[, 1])~; f = setminus(Set(vp), Set(prf)); prf = concat(prf, f); print1(vecmin(Vec(f)), ", "); ); }

A059891 a(n) = |{m : multiplicative order of 9 mod m = n}|.

Original entry on oeis.org

4, 6, 12, 14, 20, 58, 12, 88, 112, 150, 60, 290, 12, 138, 732, 144, 124, 1088, 60, 670, 740, 570, 28, 13864, 360, 138, 3968, 1362, 252, 22058, 124, 320, 1972, 1146, 732, 10704, 124, 570, 12260, 15176, 124, 60470, 28, 11634, 195728, 282, 508, 116592, 2032

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(9).
Also, number of primitive factors of 9^n - 1. - Max Alekseyev, May 03 2022

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

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), A059890 (b=8), this sequence (b=9), A059892 (b=10).
Cf. A000005, A008683, A027381, A053452, A057952, A058946, A274909.
Column k=9 of A212957.

with(numtheory):
a:= n-> add(mobius(n/d)*tau(9^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, 9^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(9^d-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = Sum_{d|n} mu(n/d)*tau(9^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).

A366577 Number of divisors of 3^n+1.

Original entry on oeis.org

2, 3, 4, 6, 4, 6, 8, 6, 8, 24, 12, 12, 8, 6, 16, 48, 4, 24, 16, 12, 8, 72, 16, 6, 64, 24, 16, 96, 16, 24, 48, 12, 4, 96, 16, 24, 16, 24, 16, 192, 32, 12, 128, 6, 32, 768, 16, 24, 16, 24, 128, 384, 16, 12, 32, 96, 64, 192, 16, 12, 128, 12, 32, 4608, 4, 24, 64

Offset: 0

Sean A. Irvine, Oct 13 2023

nonn

			a(4)=4 because 3^4+1 has divisors {1, 2, 41, 82}.

Max Alekseyev, Table of n, a(n) for n = 0..691

Cf. A000005, A002592, A034472, A046798, A057935, A057941, A074476, A274909, A366575, A366578, A366579, A366580

a:=n->numtheory[tau](3^n+1):
seq(a(n), n=0..100);

DivisorSigma[0,3^Range[0,100]+1] (* Paolo Xausa, Oct 15 2023 *)

a(n) = numdiv(3^n+1); \\ Michel Marcus, Oct 14 2023

a(n) = sigma0(3^n+1) = A000005(A034472(n)).

A366578 Sum of the divisors of 3^n+1.

Original entry on oeis.org

3, 7, 18, 56, 126, 434, 1332, 3836, 10476, 42560, 109926, 315112, 816732, 2790074, 8906760, 30220288, 64570086, 229156928, 706911048, 2034690952, 5357742012, 21838961760, 56496274632, 164750562956, 456919958880, 1517043139136, 4661686010664, 16489453890560

Offset: 0

Sean A. Irvine, Oct 13 2023

nonn

			a(4)=126 because 3^4+1 has divisors {1, 2, 41, 82}.

Max Alekseyev, Table of n, a(n) for n = 0..691

Cf. A000203, A002592, A024023, A057935, A057941, A074476, A075708, A274909, A366576, A366577, A366579, A366580

a:=n->numtheory[sigma](3^n+1):
seq(a(n), n=0..100);

DivisorSigma[1,3^Range[0,30]+1] (* Paolo Xausa, Oct 15 2023 *)

a(n) = sigma(3^n+1) = A000203(A034472(n)).

A002592 Largest prime factor of 9^n + 1.

Original entry on oeis.org

2, 5, 41, 73, 193, 1181, 6481, 16493, 21523361, 530713, 42521761, 570461, 769, 4795973261, 647753, 47763361, 926510094425921, 1743831169, 282429005041, 25480398173, 128653413121, 109688713, 56625998353, 70601370627701

Offset: 0

N. J. A. Sloane

nonn

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 2, p. 89.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Table of n, a(n) for n = 0..345
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A006530, A062396, A274909.

Cf. similar sequences listed in A274903.

[Maximum(PrimeDivisors(9^n+1)): n in [0..40]]; // Vincenzo Librandi, Jul 12 2016

for n from 0 to 30 do t1:=ifactor(9^n+1); od;

Table[FactorInteger[9^n + 1][[-1, 1]], {n, 0, 10}] (* Vincenzo Librandi, Jul 12 2016 *)

a(n) = A006530(A062396(n)). - Vincenzo Librandi, Jul 12 2016

a(n) = A074476(2*n). - Max Alekseyev, Apr 25 2022

Terms up to a(315) in b-file from Sean A. Irvine, Apr 20 2014

Terms a(316)-a(345) in b-file from Max Alekseyev, Apr 24 2019, Sep 10 2020, Aug 26 2021, Apr 25 2022

cp's OEIS Frontend

A057958 Number of prime factors of 3^n - 1 (counted with multiplicity).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

Extensions

A057952 Number of prime factors of 9^n - 1 (counted with multiplicity).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A274906 Largest prime factor of 4^n - 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A366579 a(n) = phi(3^n+1), where phi is Euler's totient function (A000010).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A366580 Number of distinct prime divisors of 3^n + 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A379642 Smallest primitive prime factor of 9^n-1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A059891 a(n) = |{m : multiplicative order of 9 mod m = n}|.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A366577 Number of divisors of 3^n+1.

Views

Author

Keywords

Examples

Links

Crossrefs