A002592 -id:A002592 - cp's OEIS frontend

A057935 Number of prime factors of 9^n + 1 (counted with multiplicity).

2, 2, 3, 3, 4, 3, 4, 2, 4, 3, 4, 6, 4, 4, 6, 2, 4, 4, 4, 5, 7, 5, 4, 4, 8, 4, 5, 6, 4, 7, 5, 2, 6, 5, 9, 8, 5, 6, 7, 5, 5, 10, 7, 6, 9, 4, 4, 6, 9, 6, 8, 7, 6, 9, 8, 9, 9, 5, 3, 11, 6, 4, 11, 6, 8, 9, 9, 8, 6, 9, 5, 6, 6, 6, 13, 4, 8, 7, 5, 4, 7, 6, 5, 11, 8, 5, 8, 7, 4, 11, 7, 9, 9, 5, 9, 7, 5, 6, 10, 7, 6

Offset: 1

Patrick De Geest, Oct 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..345 (first 329 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n+1): A057934 (b=10), this sequence (b=9), A057936 (b=8), A057937 (b=7), A057938 (b=6), A057939 (b=5), A057940 (b=4), A057941 (b=3), A054992 (b=2).

Cf. A002592, A046053, A001222, A057941, A057952, A062396.

f:=func; [f(9^n + 1):n in [1..100]]; // Marius A. Burtea, Feb 02 2020

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

a(n) = A057952(2n) - A057952(n). - T. D. Noe, Jun 19 2003

a(n) = A001222(A062396(n)) = A057941(2*n). - Amiram Eldar, Feb 02 2020

A274903 Largest prime factor of 4^n + 1.

Original entry on oeis.org

2, 5, 17, 13, 257, 41, 241, 113, 65537, 109, 61681, 2113, 673, 1613, 15790321, 1321, 6700417, 26317, 38737, 525313, 4278255361, 14449, 2931542417, 30269, 22253377, 268501, 308761441, 279073, 54410972897, 536903681, 4562284561, 384773, 67280421310721

Offset: 0

Vincenzo Librandi, Jul 11 2016

nonn

			4^3 + 1 = 65 = 5*13, so a(3) = 13.

Max Alekseyev, Table of n, a(n) for n = 0..583
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

Cf. largest prime factor of k^n+1: A002587 (k=2), A074476 (k=3), this sequence (k=4), A074478 (k=5), A274904 (k=6), A227575 (k=7), A274905 (k=8), A002592 (k=9), A003021 (k=10), A062308 (k=11).

Cf. A006530, A052539.

[Maximum(PrimeDivisors(4^n+1)): n in [0..35]];

Table[FactorInteger[4^n + 1][[-1, 1]], {n, 0, 30}]

a(n)=my(f=factor(4^n+1)[,1]); f[#f] \\ Charles R Greathouse IV, Jul 12 2016

a(n) = A006530(A052539(n)). - Michel Marcus, Jul 11 2016
a(2n) = A002590(n). a(2n+1) = A229747(n). - R. J. Mathar, Feb 28 2018
a(n) = A002587(2*n). - Amiram Eldar, Feb 01 2020

Terms to a(100) in b-file from  Vincenzo Librandi, Jul 12 2016
a(101)-a(531) in b-file from Amiram Eldar, Feb 01 2020
a(532)-a(583) in b-file from Max Alekseyev, Apr 25 2022, 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)).

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)).

A366720 Largest prime factor of 12^n+1.

Original entry on oeis.org

2, 13, 29, 19, 233, 19141, 20593, 13063, 260753, 1801, 85403261, 57154490053, 2227777, 222379, 13156924369, 35671, 1200913648289, 66900193189411, 122138321401, 905265296671, 67657441, 1885339, 68368660537, 49489630860836437, 592734049, 438472201

Offset: 0

Sean A. Irvine, Oct 17 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..306

Cf. A178248, A006530, A002587, A074476, A274903, A074478, A274904, A227575, A274905, A002592, A003021, A062308, A002590, A366712, A366713, A366714, A366715, A366716, A366717, A366718, A366719, A324941.

Table[FactorInteger[12^n + 1][[-1, 1]], {n, 0, 20}]

a(n) = A006530(A178248(n)). - Paul F. Marrero Romero, Dec 07 2023

A063271 Largest prime factor of 9^(2n)+1 (A063270).

Original entry on oeis.org

2, 41, 193, 6481, 21523361, 42521761, 769, 647753, 926510094425921, 282429005041, 128653413121, 56625998353, 24127552321, 37644053098601, 36214795668330833, 42521761, 1716841910146256242328924544641, 3833564416504313, 56227703611393, 278733912072436804273

Offset: 0

Jason Earls, Jul 12 2001

nonn

Daniel Suteu and Harry J. Smith, Table of n, a(n) for n = 0..172 (terms a(0)..a(50) from Harry J. Smith)

Cf. A006530, A063270, A074476, A002592.

Table[FactorInteger[9^(2n)+1][[-1,1]],{n,0,20}] (* Harvey P. Dale, Jan 07 2013 *)

a(n)={vecmax(factor(9^(2*n) + 1)[,1])} \\ Harry J. Smith, Aug 20 2009

a(n) = A006530(A063270(n)) = A002592(2*n) = A074476(4*n). - Daniel Suteu, May 26 2022

Definition corrected by Harry J. Smith, Aug 20 2009

A324941 Largest prime factor of 17^n + 1.

Original entry on oeis.org

2, 3, 29, 13, 41761, 101, 83233, 22796593, 184417, 5653, 63541, 87415373, 72337, 2001793, 100688449, 238212511, 52548582913, 45957792327018709121, 382069, 20352763, 1186844128302568601, 88109799136087, 6901823633, 1109309383381084655697725873, 48661191868691111041

Offset: 0

Vincenzo Librandi, Apr 05 2019

nonn

Max Alekseyev, Table of n, a(n) for n = 0..211

Cf. A006530, A224384.

Cf. A002587, A074476, A274903, A074478, A274904, A227575, A274905, A002592, A003021, A062308, A002590.

[Maximum(PrimeDivisors(17^n + 1)): n in [0..40]];

Table[FactorInteger[17^n + 1] [[-1,1]], {n, 0, 30}]

a(n) = vecmax(factor(17^n+1)[, 1]); \\ Jinyuan Wang, Apr 05 2019

a(n) = A006530(A224384(n)).

cp's OEIS Frontend

A057935 Number of prime factors of 9^n + 1 (counted with multiplicity).

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Magma

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A274903 Largest prime factor of 4^n + 1.

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Magma

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

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

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A366577 Number of divisors of 3^n+1.

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Maple

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

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Maple

Mathematica

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A366720 Largest prime factor of 12^n+1.

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Mathematica

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A063271 Largest prime factor of 9^(2n)+1 (A063270).

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