A074476 -id:A074476 - cp's OEIS frontend

A057941 Number of prime factors of 3^n + 1 (counted with multiplicity).

2, 2, 3, 2, 3, 3, 3, 3, 5, 4, 4, 3, 3, 4, 6, 2, 5, 4, 4, 3, 7, 4, 3, 6, 5, 4, 7, 4, 5, 6, 4, 2, 7, 4, 5, 4, 5, 4, 8, 5, 4, 7, 3, 5, 10, 4, 5, 4, 5, 8, 9, 4, 4, 5, 7, 6, 8, 4, 4, 7, 4, 5, 13, 2, 5, 6, 4, 5, 9, 9, 7, 8, 4, 5, 12, 6, 6, 7, 5, 5, 12, 5, 6, 10, 9, 7, 11, 6, 5, 9, 8, 4, 9, 4, 8, 6, 5, 9, 14, 6, 4

Offset: 1

Patrick De Geest, Oct 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..691 (first 658 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n+1): A057934 (b=10), A057935 (b=9), A057936 (b=8), A057937 (b=7), A057938 (b=6), A057939 (b=5), A057940 (b=4), this sequence (b=3), A054992 (b=2).

Cf. A001222, A007658, A034472, A057958, A074476.

PrimeOmega[3^Range[110]+1] (* Harvey P. Dale, Jun 20 2015 *)

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

a(n) = A057958(2n) - A057958(n) - T. D. Noe, Jun 19 2003

a(n) = A001222(A034472(n)). - Amiram Eldar, Feb 01 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

A274909 Largest prime factor of 9^n - 1.

Original entry on oeis.org

2, 5, 13, 41, 61, 73, 1093, 193, 757, 1181, 3851, 6481, 797161, 16493, 4561, 21523361, 34511, 530713, 363889, 42521761, 368089, 570461, 23535794707, 6481, 22996651, 4795973261, 19927, 647753, 20381027, 47763361, 22434744889, 926510094425921, 2413941289

Offset: 1

Vincenzo Librandi, Jul 11 2016

nonn

			9^4 - 1 = 6560 = 2^5*5*41, so a(4) = 41.

Table of n, a(n) for n = 1..691
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

Cf. A006530, A024101, A074477.

Cf. similar sequences listed in A274906.

[Maximum(PrimeDivisors(9^n-1)): n in [1..40]];

Table[FactorInteger[9^n-1][[-1,1]],{n,40}]

a(n) = A006530(A024101(n)).
a(n) = A074477(2*n). - Amiram Eldar, Feb 02 2020
a(n) = max(A074476(n),A074477(n)). - Max Alekseyev, Apr 25 2022

Terms to a(100) in b-file from Vincenzo Librandi, Jul 13 2016
a(101)-a(330) in b-file from Amiram Eldar, Feb 02 2020
a(331)-a(691) in b-file from Max Alekseyev, May 22 2022, Jul 25 2023

A074477 Largest prime factor of 3^n - 1.

Original entry on oeis.org

2, 2, 13, 5, 11, 13, 1093, 41, 757, 61, 3851, 73, 797161, 1093, 4561, 193, 34511, 757, 363889, 1181, 368089, 3851, 1001523179, 6481, 391151, 797161, 8209, 16493, 20381027, 4561, 4404047, 21523361, 2413941289, 34511, 2664097031, 530713

Offset: 1

Rick L. Shepherd, Aug 23 2002

nonn

			3^7 - 1 = 2186 = 2*1093, so a(7) = 1093.

Table of n, a(n) for n = 1..690
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A006530 (largest prime factor), A024023 (3^n-1).

Cf. A074476 (largest prime factor of 3^n + 1), A005420 (largest prime factor of 2^n - 1), A074479 (largest prime factor of 5^n - 1).

[Maximum(PrimeDivisors(3^n-1)): n in [1..40]]; // Vincenzo Librandi, Aug 23 2013

A074477 := proc(n)
        A006530( 3^n-1) ;
end proc: # R. J. Mathar, Jul 18 2015
# alternative:
a:= n-> max(seq(i[1], i=ifactors(3^n-1)[2])):
seq(a(n), n=1..40);  # Alois P. Heinz, Jul 18 2015

Table[FactorInteger[3^n - 1] [[-1, 1]], {n, 40}] (* Vincenzo Librandi, Aug 23 2013 *)

for(n=1,40, v=factor(3^n-1); print1(v[matsize(v)[1],1],","))

a(n) = A006530(A024023(n)). - Michel Marcus, Jul 18 2015

Terms to a(100) in b-file from Vincenzo Librandi, Aug 23 2013
a(101)-a(660) in b-file from Amiram Eldar, Feb 01 2020
a(661)-a(690) in b-file from Max Alekseyev, May 22 2022

A074478 Largest prime factor of 5^n + 1.

Original entry on oeis.org

2, 3, 13, 7, 313, 521, 601, 449, 11489, 5167, 9161, 5281, 390001, 38923, 234750601, 7621, 29423041, 41540861, 6597973, 213029, 632133361, 7603, 1030330938209, 42272797713043, 152587500001, 50150933101, 83181652304609, 16018507

Offset: 0

Rick L. Shepherd, Aug 23 2002

nonn

			5^11 + 1 = 48828126 = 2*3*23*67*5281, so a(11) = 5281.

Table of n, a(n) for n = 0..471
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A002587 (largest prime factor of 2^n + 1), A074479 (largest prime factor of 5^n - 1), A074476 (largest prime factor of 3^n + 1), A227575 (largest prime factor of 7^n + 1).

Cf. A006530, A034474.

[Maximum(PrimeDivisors(5^n+1)): n in [0..30]]; // Vincenzo Librandi, Jul 09 2016

Table[FactorInteger[5^n + 1][[-1, 1]], {n, 0, 30}] (* Bruno Berselli, Aug 23 2013 *)

for(n=0,30, v=factor(5^n+1); print1(v[matsize(v)[1],1],","))

a(n) = A006530(A034474(n)). - Michel Marcus, Jul 09 2016

Terms to a(100) in b-file from Vincenzo Librandi, Jul 09 2016
a(101)-a(451) in b-file from Amiram Eldar, Feb 01 2020
a(452)-a(471) in b-file from Max Alekseyev, Apr 25 2022, Jan 04 2024

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)).

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

A057941 Number of prime factors of 3^n + 1 (counted with multiplicity).

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A274903 Largest prime factor of 4^n + 1.

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Magma

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A274909 Largest prime factor of 9^n - 1.

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Magma

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A074477 Largest prime factor of 3^n - 1.

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A074478 Largest prime factor of 5^n + 1.

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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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