A005420 -id:A005420 - cp's OEIS frontend

A086251 Number of primitive prime factors of 2^n - 1.

0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 2, 1, 2, 3, 3, 3, 1, 3, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 3, 1, 2, 3, 2, 3, 2, 2, 3, 1, 1, 3, 1, 3, 2, 2, 2, 1, 1, 2, 2, 1, 1, 3, 4, 1, 2, 3, 2, 2, 1, 3, 3, 2, 3, 2, 2, 3

Offset: 1

T. D. Noe, Jul 14 2003

A prime factor of 2^n - 1 is called primitive if it does not divide 2^r - 1 for any r < n. Equivalently, p is a primitive prime factor of 2^n - 1 if ord(2,p) = n. Zsigmondy's theorem says that there is at least one primitive prime factor for n > 1, except for n=6. See A086252 for those n that have a record number of primitive prime factors.
Number of odd primes p such that A002326((p-1)/2) = n. Number of occurrences of number n in A014664. - Thomas Ordowski, Sep 12 2017
The prime factors are not counted with multiplicity, which matters for a(364)=4 and a(1755)=6. - Jeppe Stig Nielsen, Sep 01 2020

			a(11) = 2 because 2^11 - 1 = 23*89 and both 23 and 89 have order 11.

Table of n, a(n) for n = 1..1206
Eric Weisstein's World of Mathematics, Zsigmondy Theorem

Cf. A046800, A046051 (number of prime factors, with repetition, of 2^n-1), A086252, A002588, A005420, A002184, A046801, A049093, A049094, A059499, A085021, A097406, A112927, A237043.

Join[{0}, Table[cnt=0; f=Transpose[FactorInteger[2^n-1]][[1]]; Do[If[MultiplicativeOrder[2, f[[i]]]==n, cnt++ ], {i, Length[f]}]; cnt, {n, 2, 200}]]

a(n) = sumdiv(n, d, moebius(n/d)*omega(2^d-1)); \\ Michel Marcus, Sep 12 2017

a(n) = my(m=polcyclo(n, 2)); omega(m/gcd(m,n)) \\ Jeppe Stig Nielsen, Sep 01 2020

a(n) = Sum{d|n} mu(n/d) A046800(d), inverse Mobius transform of A046800.
a(n) <= A182590(n). - Thomas Ordowski, Sep 14 2017
a(n) = A001221(A064078(n)). - Thomas Ordowski, Oct 26 2017

Terms to a(500) in b-file from T. D. Noe, Nov 11 2010
Terms a(501)-a(1200) in b-file from Charles R Greathouse IV, Sep 14 2017
Terms a(1201)-a(1206) in b-file from Max Alekseyev, Sep 11 2022

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

A063670 Positions of nonzero coefficients in cyclotomic polynomial Phi_n(x), converted from binary to decimal.

Original entry on oeis.org

2, 3, 3, 7, 5, 31, 7, 127, 17, 73, 31, 2047, 21, 8191, 127, 443, 257, 131071, 73, 524287, 341, 7003, 2047, 8388607, 273, 1082401, 8191, 262657, 5461, 536870911, 443, 2147483647, 65537, 1797851, 131071, 26181091, 4161, 137438953471, 524287

Offset: 0

Antti Karttunen, Aug 03 2001

a(n) = 2^n-1 whenever n is prime. It seems as if a(n) >= A005420(n) for all n (checked up to 200), with equality for all 1A005420(n)=2^n-1 (i.e., 2^n-1 is prime). - M. F. Hasler, Apr 30 2007

a(0) could also be 1. - T. D. Noe, Oct 29 2007

Cf. A013594.
a(n) = A063696(n) (the positive terms) + A063698(n) (the negative terms).
This sequence in binary: A063671.
Cf. A005420.

[seq(Phi_pos_terms(j,2)+Phi_neg_terms(j,2),j=0..104)];

a[n_] := FromDigits[ If[# != 0, 1, 0]& /@ CoefficientList[ Cyclotomic[n, x], x], 2]; a[0] = 2; Table[a[n], {n, 0, 38}] (* Jean-François Alcover, Dec 11 2012 *)

A063670(n)=local(p=polcyclo(n+!n)); if(n,sum(i=0, n, (polcoeff(p, i)<>0)<M. F. Hasler, Apr 30 2007

a(n) = subst(apply(x->x!=0, polcyclo(n,'x)), 'x, 2);  \\ Gheorghe Coserea, Nov 04 2016

A366718 Largest prime factor of 12^n - 1.

Original entry on oeis.org

11, 13, 157, 29, 22621, 157, 4943, 233, 80749, 22621, 266981089, 20593, 20369233, 13063, 22621, 260753, 74876782031, 80749, 29043636306420266077, 85403261, 8177824843189, 57154490053, 321218438243, 2227777, 12629757106815551, 20369233, 86769286104133

Offset: 1

Sean A. Irvine, Oct 17 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..310
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

Cf. A024140, A006530, A005420, A074477, A274906, A074479, A274907, A074249, A274908, A274909, A005422, A274910, A366707, A366708, A366709, A366710, A366711, A366717, A366720.

[Maximum(PrimeDivisors(12^n-1)): n in [1..40]];

Table[FactorInteger[12^n - 1][[-1, 1]], {n, 40}]

a(n) = A006530(A024140(n)).

A367003 a(n) is the largest prime factor of n*2^n-1.

Original entry on oeis.org

1, 7, 23, 7, 53, 383, 179, 89, 271, 3413, 2503, 2137, 59, 367, 1433, 41, 15803, 59729, 26423, 11161, 1559, 12611, 9187523, 127867, 119837257, 11527, 2360833, 43969, 2339, 32212254719, 257503, 616318177, 260587, 127873, 682902239, 44939, 69660839431, 1185617

Offset: 1

Sean A. Irvine, Oct 31 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 1..865

Cf. A003261, A006530, A005420, A367002, A367004, A366899.

a[n_] := FactorInteger[n*2^n - 1][[-1, 1]]; Array[a, 40] (* Amiram Eldar, Oct 29 2024 *)

a(n) = A006530(A003261(n)).

A002588 a(n) = largest noncomposite factor of 2^(2n+1) - 1.

Original entry on oeis.org

1, 7, 31, 127, 73, 89, 8191, 151, 131071, 524287, 337, 178481, 1801, 262657, 2089, 2147483647, 599479, 122921, 616318177, 121369, 164511353, 2099863, 23311, 13264529, 4432676798593, 131071, 20394401, 201961, 1212847, 3203431780337

Offset: 0

N. J. A. Sloane

nonn

a(n) is also the largest noncomposite factor of A147590(n). - César Aguilera, Jul 31 2019

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. 84.
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).

Max Alekseyev, Table of n, a(n) for n = 0..602 (terms 1..494 from Sean A. Irvine)
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. A002184, A005420, A147590.

[1] cat [Maximum(PrimeDivisors(2^(2*n+1) - 1)): n in [1..60]]; // Vincenzo Librandi, Aug 02 2019

Table[FactorInteger[2^(2 n + 1) - 1] [[-1, 1]], {n, 0, 30}] (* Vincenzo Librandi, Aug 02 2019 *)

a(n) = if (n==0, 1, vecmax(factor(2^(2*n+1) - 1)[, 1])); \\ Michel Marcus, Aug 03 2019

More terms from Don Reble, Nov 14 2006

A097407 a(n) = (A097406(n) - 1)/n.

Original entry on oeis.org

0, 1, 2, 1, 6, 0, 18, 2, 8, 1, 8, 1, 630, 3, 10, 16, 7710, 1, 27594, 2, 16, 31, 7760, 10, 72, 105, 9728, 4, 72, 11, 69273666, 2048, 18166, 1285, 3512, 3, 16657248, 4599, 3112, 1542, 4012472, 129, 48834, 48, 518, 60787, 282224, 14, 90462791808, 81, 218, 31

Offset: 1

Marco Matosic, Aug 16 2004

nonn

Bill McEachen, Table of n, a(n) for n = 1..1206
S. S. Wagstaff, Jr., The Cunningham Project, Factorizations of 2^n-1, n odd, n<1200.

Cf. A097406, A005420, A186283.

Edited by Vladeta Jovovic, Aug 26 2004

A336719 Largest odd prime p for which the order of 2 mod p is at most n.

Original entry on oeis.org

3, 7, 7, 31, 31, 127, 127, 127, 127, 127, 127, 8191, 8191, 8191, 8191, 131071, 131071, 524287, 524287, 524287, 524287, 524287, 524287, 524287, 524287, 524287, 524287, 524287, 524287, 2147483647, 2147483647, 2147483647, 2147483647, 2147483647, 2147483647

Offset: 2

Jeppe Stig Nielsen, Aug 01 2020

nonn

a(1) is undefined.
Changing "at most n" to "equal to n" in the definition gives A097406.
The first term that is not a Mersenne prime (A000668) is 4432676798593.
For a version without duplicates, see A336720. For a list of all n where a(n) increases, see A336721.

Jeppe Stig Nielsen, Table of n, a(n) for n = 2..251

Cf. A000668, A005420, A014664, A097406, A336720, A336721.

re=0;for(n=2,+oo,p=vecmax(factor(2^n-1)[,1]);p>re&&re=p;print1(re,", "))

A367005 a(n) is the largest prime factor of n*2^n+1 for n>0, and a(0)=1.

Original entry on oeis.org

1, 3, 3, 5, 13, 23, 11, 23, 683, 419, 19, 1733, 199, 11833, 487, 997, 61681, 4691, 211, 5279, 7541, 1914791, 7177, 607, 5233, 6689, 2373919, 336823, 8937209, 6051013, 409, 11681, 25781083, 6031230671, 18803, 32502455213, 934861, 339016085231, 55586743

Offset: 0

Sean A. Irvine, Oct 31 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 0..865

Cf. A002064, A006530, A005420, A367003, A367004.

A367005[n_]:=FactorInteger[n*2^n+1][[-1,1]];Array[A367005,50,0] (* Paolo Xausa, Nov 09 2023 *)

a(n) = A006530(A002064(n)).

Name edited by Michel Marcus, Nov 10 2023

A249780 Product of lowest and highest prime factors of 2^n-1.

Original entry on oeis.org

9, 49, 15, 961, 21, 16129, 51, 511, 93, 2047, 39, 67092481, 381, 1057, 771, 17179607041, 219, 274876858369, 123, 2359, 2049, 8388607, 723, 55831, 24573, 1838599, 381, 486737, 993, 4611686014132420609, 196611, 4196353, 393213, 3810551, 327, 137438953471, 1572861, 849583, 185043

Offset: 2

Jacob Vecht, Nov 05 2014

nonn

			The lowest and higest prime factors of 2^6-1 are 3 and 7, so A(6) = 21

Chai Wah Wu, Table of n, a(n) for n = 2..200

a:= proc(n) local F; F:= numtheory:-factorset(2^n-1); min(F)*max(F) end proc:
seq(a(n),n=2..50); # Robert Israel, Nov 05 2014

plhpf[n_]:=Module[{fn=FactorInteger[n]},fn[[1,1]]fn[[-1,1]]]; Table[plhpf [2^n-1],{n,2,40}] (* Harvey P. Dale, May 23 2020 *)

for(n=2, 50, p=2^n-1; print1(factor(p)[1, 1]*factor(p)[#factor(p)[, 1], 1], ", ")) \\ Derek Orr, Nov 05 2014

from sympy import primefactors
A249780_list, x = [], 1
for n in range(2,10):
    x = 2*x + 1
    p = primefactors(x)
    A249780_list.append(max(p)*min(p)) # Chai Wah Wu, Nov 05 2014

a(n) = A005420(n) * A049479(n)

More terms from Derek Orr, Nov 05 2014

cp's OEIS Frontend

A086251 Number of primitive prime factors of 2^n - 1.

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A274908 Largest prime factor of 8^n - 1.

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A063670 Positions of nonzero coefficients in cyclotomic polynomial Phi_n(x), converted from binary to decimal.

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A366718 Largest prime factor of 12^n - 1.

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A367003 a(n) is the largest prime factor of n*2^n-1.

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A002588 a(n) = largest noncomposite factor of 2^(2n+1) - 1.

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A097407 a(n) = (A097406(n) - 1)/n.

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A336719 Largest odd prime p for which the order of 2 mod p is at most n.