A274906 -id:A274906 - cp's OEIS frontend

A005420 Largest prime factor of 2^n - 1.

3, 7, 5, 31, 7, 127, 17, 73, 31, 89, 13, 8191, 127, 151, 257, 131071, 73, 524287, 41, 337, 683, 178481, 241, 1801, 8191, 262657, 127, 2089, 331, 2147483647, 65537, 599479, 131071, 122921, 109, 616318177, 524287, 121369, 61681, 164511353, 5419

Offset: 2

N. J. A. Sloane

nonn

			2^6 - 1 = 63 = 3*21 = 9*7, so a(6) = 7.

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Charles R Greathouse IV and Amiram Eldar, Table of n, a(n) for n = 2..1206 (terms up to 500 from T. D. Noe, terms 501..1000 from Charles R Greathouse IV, terms 1001..1206 from Amiram Eldar)
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
factordb.com, Status of 2^1207-1. The factorization of the composite factor C337 of 2^1207-1 with 337 decimal digits is considered by many to be the most desired open factorization problem.
R. K. Guy, Letter to G. B. Huff & N. J. A. Sloane, Aug 1974
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Mersenne Number

Cf. A000043, A000225, A006530, A002326, A002587.
Cf. similar sequences listed in A274906.
Cf. A337431 (a(n)=a(2n)), A359063 (a(n)=a(2n)=a(4n)), A359088.

[Maximum(PrimeDivisors(2^n-1)): n in [2..45]]; // Vincenzo Librandi, Jul 13 2016

a[n_] := a[n] = FactorInteger[2^n-1] // Last // First; Table[Print[{n, a[n]}, If[2^n-1 == a[n], " Mersenne prime", " "]]; a[n], {n, 2, 127}] (* Jean-François Alcover, Dec 11 2012 *)
Table[FactorInteger[2^n - 1][[-1, 1]], {n, 2, 40}] (* Vincenzo Librandi, Jul 13 2016 *)

for(n=2,44, v=factor(2^n-1)[,1]; print1(v[#v]", "));

a(n) = vecmax(factor(2^n-1)[,1]); \\ Michel Marcus, Dec 15 2022

a(n) = a(2n) iff a(n) > A002587(n). See A337431. - Thomas Ordowski, Jan 07 2014
a(n) = A006530(A000225(n)). - Vincenzo Librandi, Jul 13 2016
a(n) = 2^n-1 = A000225(n) iff n is a Mersenne exponent (A000043). - Bernard Schott, Dec 11 2022

Description corrected by Michael Somos, Feb 24 2002
More terms from Rick L. Shepherd, Aug 22 2002
Incorrect comments removed by Michel Marcus, Dec 15 2022

A046798 Number of divisors of 2^n + 1.

Original entry on oeis.org

2, 2, 2, 3, 2, 4, 4, 4, 2, 8, 6, 4, 4, 4, 8, 12, 2, 4, 16, 4, 4, 12, 8, 4, 8, 16, 16, 20, 4, 8, 48, 4, 4, 24, 16, 32, 16, 8, 16, 12, 4, 8, 64, 4, 8, 64, 32, 8, 8, 8, 64, 48, 8, 8, 64, 48, 8, 24, 8, 16, 16, 4, 32, 64, 4, 64, 64, 8, 12, 24, 96, 8, 32, 8, 32, 96, 16, 64, 768, 4, 8, 192, 32, 64

Offset: 0

Labos Elemer

nonn

a(n) is odd iff n = 3, as a consequence of the Catalan-Mihăilescu theorem. - Bernard Schott, Oct 05 2021

			a(7)=4, because 2^7 + 1 = 129 has 4 divisors.

Max Alekseyev, Table of n, a(n) for n = 0..1128 (terms 0..500 from T. D. Noe, terms 501..1062 from Amiram Eldar, term 1108 from Tyler Busby)
Wikipedia, Catalan's conjecture.

Cf. A000005, A000051, A002587, A002589, A046801, A054992, A057957, A059886, A274906, A344897.

[NumberOfDivisors(2^n+1): n in [0..100]]; // Vincenzo Librandi, Feb 05 2018

a:= n-> numtheory[tau](2^n+1):
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 23 2021

A046798[n_IntegerQ]:=DivisorSigma[0,1+2^n]; (* Enrique Pérez Herrero, Nov 09 2010 *)
DivisorSigma[0, 1 + 2^#] & /@ Range[0, 83] (* Jayanta Basu, Jun 29 2013 *)
Table[DivisorSigma[0, 2^n + 1], {n, 0, 100}] (* Vincenzo Librandi, Feb 05 2018 *)

a(n) = numdiv(2^n+1); \\ Michel Marcus, Mar 18 2017

from sympy.ntheory import divisor_count
def A046798(n): return divisor_count(2**n + 1) # Indranil Ghosh, Mar 18 2017

a(n) = A000005(A000051(n)). - Michel Marcus, Mar 18 2017

A057957 Number of prime factors of 4^n - 1 (counted with multiplicity).

Original entry on oeis.org

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

Offset: 1

Patrick De Geest, Nov 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..1128 (first 603 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), this sequence (b=4), A057958 (b=3), A046051 (b=2).

Cf. A001222, A024036, A054992, A085029, A274906.

PrimeOmega/@(4^Range[90]-1) (* Harvey P. Dale, Dec 31 2018 *)

Mobius transform of A085029. - T. D. Noe, Jun 19 2003

a(n) = A001222(A024036(n)) = A046051(2*n). - Amiram Eldar, Feb 01 2020

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

Original entry on oeis.org

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).

A005422 Largest prime factor of 10^n - 1.

Original entry on oeis.org

3, 11, 37, 101, 271, 37, 4649, 137, 333667, 9091, 513239, 9901, 265371653, 909091, 2906161, 5882353, 5363222357, 333667, 1111111111111111111, 27961, 10838689, 513239, 11111111111111111111111, 99990001, 182521213001, 1058313049

Offset: 1

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.
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 = 1..352
K. Beschorner, Factorizations of Repunit Numbers of the form (10^n-1)/9 (111111...). [Retrieved July 30, 2015]
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Yousuke Koide, Factorizations of Repunit Numbers.
S. S. Wagstaff, Jr., The Cunningham Project

Same as A003020 except for the additional a(1) = 3.
Cf. A002275, A002283, A004023, A006530, A067063, A075024, A102380.
Cf. similar sequences listed in A274906.

[Maximum(PrimeDivisors(10^n-1)): n in [1..45]]; // Vincenzo Librandi, Jul 13 2016

A005422 := proc(n)
    10^n-1 ;
    A006530(%) ;
end proc: # R. J. Mathar, Dec 02 2016

Table[FactorInteger[10^n - 1][[-1, 1]], {n, 1, 40}] (* Vincenzo Librandi, Jul 13 2016 *)

a(n)=vecmax(factor(10^n-1)[,1]) \\ Simplified by M. F. Hasler, Jul 30 2015

For n > 1, a(n) = A003020(n). For 1 < n < 10, a(n) = A075024(n). - M. F. Hasler, Jul 30 2015
a(n) = A006530(A002283(n)). - Vincenzo Librandi, Jul 13 2016
a(A004023(n)) = A002275(A004023(n)). - Bernard Schott, May 24 2022

Terms to a(100) in b-file from Yousuke Koide added by T. D. Noe, Dec 06 2006
Edited by M. F. Hasler, Jul 30 2015
a(101)-a(322) in b-file from Ray Chandler, Apr 22 2017
a(323)-a(352) in b-file from Max Alekseyev, Apr 26 2022

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

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

A074479 Largest prime factor of 5^n - 1.

Original entry on oeis.org

2, 3, 31, 13, 71, 31, 19531, 313, 829, 521, 12207031, 601, 305175781, 19531, 1741, 11489, 466344409, 5167, 3981071, 9161, 519499, 12207031, 332207361361, 390001, 9384251, 305175781, 31051, 234750601, 22125996444329, 7621

Offset: 1

Rick L. Shepherd, Aug 23 2002

nonn

			5^9 - 1 = 1953124 = (2^2)*19*31*829, so a(9) = 829.

Table of n, a(n) for n = 1..502
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A074478 (largest prime factor of 5^n + 1), A074477 (largest prime factor of 3^n - 1), A074249 (largest prime factor of 7^n - 1).
Cf. A006530, A024049.
Cf. similar sequences listed in A274906.

[Maximum(PrimeDivisors(5^n-1)): n in [1..45]]; // Vincenzo Librandi, Jul 13 2016

Table[FactorInteger[5^n - 1] [[-1, 1]], {n, 30}] (* Vincenzo Librandi, Aug 23 2013 *)

for(n=1,32, v=factor(5^n-1); print1(v[matsize(v)[1],1],","))

a(n) = A006530(A024049(n)). - Vincenzo Librandi, Jul 13 2016

Terms to a(100) in b-file from Vincenzo Librandi, Aug 23 2013
a(101)-a(448) in b-file from Amiram Eldar, Feb 01 2020
a(449)-a(502) in b-file from Max Alekseyev, Apr 25 2022

A274907 Largest prime factor of 6^n - 1.

Original entry on oeis.org

5, 7, 43, 37, 311, 43, 55987, 1297, 2467, 311, 3154757, 97, 760891, 55987, 1201, 98801, 30839, 46441, 638073026189, 6781, 1822428931, 51828151, 7505944891, 1678321, 40185601, 760891, 623067280651, 5030761, 7369130657357778596659, 1950271, 49744740983476472807

Offset: 1

Vincenzo Librandi, Jul 11 2016

nonn

			6^5 - 1 = 7775 = 5*5*311, so a(5) = 311.

Max Alekseyev, Table of n, a(n) for n = 1..430 (terms 1..100 from Vincenzo Librandi, term 101..420 from Amiram Eldar)
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

Cf. A006530, A057955, A059888, A085031, A024062, A366620, A366621, A366622, A366623.

Cf. similar sequences listed in A274906.

[Maximum(PrimeDivisors(6^n-1)): n in [1..40]];

Table[FactorInteger[6^n - 1][[-1, 1]], {n, 40}]

a(n) = vecmax(factor(6^n-1)[,1]); \\ Michel Marcus, Jul 13 2016

a(n) = A006530(A024062(n)). - Michel Marcus, Jul 11 2016

A274908 Largest prime factor of 8^n - 1.

Original entry on oeis.org

7, 7, 73, 13, 151, 73, 337, 241, 262657, 331, 599479, 109, 121369, 5419, 23311, 673, 131071, 262657, 1212847, 1321, 649657, 599479, 10052678938039, 38737, 10567201, 22366891, 97685839, 14449, 9857737155463, 18837001, 658812288653553079, 22253377

Offset: 1

Vincenzo Librandi, Jul 11 2016

nonn

			8^5 -1 = 32767 = 7*31*151, so a(5) = 151.

Table of n, a(n) for n = 1..500
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

Cf. A005420, A006530, A024088, A274905.

Cf. similar sequences listed in A274906.

[Maximum(PrimeDivisors(8^n-1)): n in [1..40]];

f:= n -> max(map(t -> max(numtheory:-factorset(subs(x=2,t[1]))), factors(x^(3*n)-1)[2])):
map(f, [$1..120]); # Robert Israel, Jul 12 2016

Table[FactorInteger[8^n - 1][[-1, 1]], {n, 40}]

a(n) = A006530(A024088(n)). - Michel Marcus, Jul 11 2016

a(n) = A005420(3*n). - Robert Israel, Jul 12 2016

Terms to a(100) in b-file from Vincenzo Librandi, Jul 13 2016
a(101)-a(402) in b-file from Amiram Eldar, Feb 02 2020
a(403)-a(500) in b-file from Max Alekseyev, Apr 25 2022, Sep 11 2022, Dec 05 2022, Feb 25 2023

cp's OEIS Frontend

A005420 Largest prime factor of 2^n - 1.

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A046798 Number of divisors of 2^n + 1.

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A057957 Number of prime factors of 4^n - 1 (counted with multiplicity).

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

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A005422 Largest prime factor of 10^n - 1.

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A069061 Sum of divisors of 2^n+1.

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A274909 Largest prime factor of 9^n - 1.

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