A274903 -id:A274903 - cp's OEIS frontend

A052539 a(n) = 4^n + 1.

2, 5, 17, 65, 257, 1025, 4097, 16385, 65537, 262145, 1048577, 4194305, 16777217, 67108865, 268435457, 1073741825, 4294967297, 17179869185, 68719476737, 274877906945, 1099511627777, 4398046511105, 17592186044417

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

The sequence is a Lucas sequence V(P,Q) with P = 5 and Q = 4, so if n is a prime number, then V_n(5,4) - 5 is divisible by n. The smallest pseudoprime q which divides V_q(5,4) - 5 is 15.
Also the edge cover number of the (n+1)-Sierpinski tetrahedron graph. - Eric W. Weisstein, Sep 20 2017
First bisection of A000051, A049332, A052531 and A014551. - Klaus Purath, Sep 23 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..175
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 470.
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences 8(10) (2019).
Eric Weisstein's World of Mathematics, Edge Cover Number.
Eric Weisstein's World of Mathematics, Sierpinski Tetrahedron Graph.
Wikipedia, Lucas sequence: Specific names.
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A164346 (first differences).
Other powers: A000051, A034472, A034474, A062394, A034491, A062395, A062396, A062397, A007689, A063376, A063481, A074600-A074624, A034524, A178248, A228081.
Cf. A019434, A274903.

List([0..30], n-> 4^n+1); # G. C. Greubel, May 09 2019

[4^n+1: n in [0..30] ]; // Vincenzo Librandi, Apr 30 2011

spec := [S,{S=Union(Sequence(Union(Z,Z,Z,Z)),Sequence(Z))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..30);
A052539:=n->4^n + 1; seq(A052539(n), n=0..30); # Wesley Ivan Hurt, Jun 12 2014

Table[4^n + 1, {n, 0, 30}]
(* From Eric W. Weisstein, Sep 20 2017 *)
4^Range[0, 30] + 1
LinearRecurrence[{5, -4}, {2, 5}, 30]
CoefficientList[Series[(2-5x)/(1-5x+4x^2), {x, 0, 30}], x] (* End *)

a(n)=4^n+1 \\ Charles R Greathouse IV, Nov 20 2011

[4^n+1 for n in (0..30)] # G. C. Greubel, May 09 2019

a(n) = 4^n + 1.
a(n) = 4*a(n-1) - 3 = 5*a(n-1) - 4*a(n-2).
G.f.: (2 - 5*x)/((1 - 4*x)*(1 - x)).
E.g.f.: exp(x) + exp(4*x). - Mohammad K. Azarian, Jan 02 2009
From Klaus Purath, Sep 23 2020: (Start)
a(n) = 3*4^(n-1) + a(n-1).
a(n) = (a(n-1)^2 + 9*4^(n-2))/a(n-2).
a(n) = A178675(n) - 3. (End)

A002587 Largest prime factor of 2^n + 1.

Original entry on oeis.org

2, 3, 5, 3, 17, 11, 13, 43, 257, 19, 41, 683, 241, 2731, 113, 331, 65537, 43691, 109, 174763, 61681, 5419, 2113, 2796203, 673, 4051, 1613, 87211, 15790321, 3033169, 1321, 715827883, 6700417, 20857, 26317, 86171, 38737, 25781083, 525313

Offset: 0

N. J. A. Sloane

nonn

a(n) != 1 (mod n) for n = 3, 51, 141, 309, 321, 348, ... - Giovanni Resta & Thomas Ordowski, Jan 05 2014

a(n) != 1 (mod n) iff a(m) = a(n) for some m < n. Then n = 3m for m = 1, 17, 47, 103, 107, 116, ... - Thomas Ordowski, Jan 08 2014

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. 85.
E. Lucas, Théorie des fonctions numériques simplement périodiques, Amer. J. Math., 1 (1878), 184-239, 289-321.
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..1128 (terms 1..500 from T. D. Noe, terms 1037..1062 from Amiram Eldar, term 1108 from Tyler Busby)
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
D. X. Charles, The abc-conjecture and the largest prime factor of 2^n + 1
Edouard Lucas, Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240, 289-321. See pages 239 and 240.
Edouard Lucas, The Theory of Simply Periodic Numerical Functions, Fibonacci Association, 1969. English translation of article "Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240.
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A000051, A002586, A006530.

Cf. similar sequences listed in A274903.

[Maximum(PrimeDivisors(2^n+1)): n in [0..40]]; // Vincenzo Librandi, Jul 12 2016

Table[FactorInteger[2^n + 1][[-1, 1]], {n, 0, 30}] (* Vincenzo Librandi, Jul 12 2016 *)

a(n)=my(f=factor(2^n+1)[,1]); f[#f] \\ Charles R Greathouse IV, Jul 12 2016

Charles proves that a(n) >> n^(4/3) infinitely often under the abc conjecture, and reports that Andrew Granville has improved this to a(n) >> n^2. - Charles R Greathouse IV, Apr 29 2013

a(n) = A006530(A000051(n)). - Vincenzo Librandi, Jul 12 2016

More terms from James Sellers, Jul 06 2000

Offset 0, a(0) = 2 from Vincenzo Librandi, Jul 12 2016

A057940 Number of prime factors of 4^n + 1 (counted with multiplicity).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 3, 1, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 3, 5, 3, 7, 3, 6, 3, 3, 4, 5, 2, 6, 4, 7, 5, 5, 4, 10, 3, 5, 5, 5, 4, 11, 2, 4, 3, 6, 6, 9, 2, 4, 6, 7, 5, 8, 3, 7, 6, 6, 4, 10, 2, 10, 7, 6, 4, 8, 4, 6, 7, 5, 2, 14, 4, 9, 5, 4, 4, 10, 4, 6, 8, 11, 4, 8, 3, 4, 8, 11, 4, 9, 5, 10, 4, 9, 8, 12, 6

Offset: 1

Patrick De Geest, Oct 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..583 (first 531 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), this sequence (b=4), A057941 (b=3), A054992 (b=2).

Cf. A001222, A052539, A057957, A274903.

with(numtheory); A057940:=n->bigomega(4^n + 1); seq(A057940(n), n=1..100); # Wesley Ivan Hurt, Jan 28 2014

Table[PrimeOmega[4^n + 1], {n, 100}] (* Wesley Ivan Hurt, Jan 28 2014 *)

a(n) = A057957(2n) - A057957(n). - T. D. Noe, Jun 19 2003
a(n) = Omega(4^n + 1) = A001222(A052539(n)). - Wesley Ivan Hurt, Jan 28 2014
a(n) = A054992(2*n). - Amiram Eldar, Feb 01 2020

A366608 a(n) = phi(4^n+1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 4, 16, 48, 256, 800, 3840, 12544, 65536, 186624, 986880, 3345408, 16515072, 52306176, 252645120, 760320000, 4288266240, 13628740608, 64258375680, 218462552064, 1095233372160, 3105655160832, 16510446886912, 56000724240384, 280012271910912, 869940000000000

Offset: 0

Sean A. Irvine, Oct 14 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..583

Cf. A000010, A052539, A057940, A274903, A295501, A366605, A366606, A366607, A366609.

Cf. A053285, A366579, A366618, A366630, A366639, A366658, A366667, A366669, A366690, A366716.

EulerPhi[4^Range[0,30]+1] (* Paolo Xausa, Oct 14 2023 *)

PARI
```
{a(n) = eulerphi(4^n+1)}
```

from sympy import totient
def A366608(n): return totient((1<<(n<<1))+1) # Chai Wah Wu, Oct 14 2023

a(n) = A053285(2*n). - Max Alekseyev, Jan 08 2024

A274905 Largest prime factor of 8^n + 1.

Original entry on oeis.org

2, 3, 13, 19, 241, 331, 109, 5419, 673, 87211, 1321, 20857, 38737, 22366891, 14449, 18837001, 22253377, 43691, 279073, 160465489, 4562284561, 77158673929, 4327489, 168749965921, 487824887233, 1133836730401, 21841, 272010961, 88959882481, 96076791871613611

Offset: 0

Vincenzo Librandi, Jul 11 2016

nonn

Table of n, a(n) for n = 0..502
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

Cf. A002587, A006530, A062395.

Cf. similar sequences listed in A274903.

[Maximum(PrimeDivisors(8^n+1)): n in [0..40]];

Maple
```
8^4 + 1 = 4097 = 17*241, so a(4) = 241.
```

Table[FactorInteger[8^n + 1][[-1, 1]], {n, 0, 40}]

a(n) = A006530(A062395(n)). - Michel Marcus, Jul 11 2016

a(n) = A002587(3*n). - Amiram Eldar, Feb 02 2020

Terms to a(100) in b-file from Vincenzo Librandi, Jul 12 2016
a(101)-a(354) in b-file from Amiram Eldar, Feb 02 2020
a(355)-a(502) in b-file from Max Alekseyev, May 28 2022, Sep 06 2022, Feb 25 2023

A366607 Sum of the divisors of 4^n+1.

Original entry on oeis.org

3, 6, 18, 84, 258, 1302, 4356, 20520, 65538, 351120, 1110276, 5048232, 17041416, 82623888, 284225796, 1494039792, 4301668356, 20788904016, 73234343952, 332019460560, 1103789883396, 5936210280000, 18679788287496, 84884999116320, 282937726148616

Offset: 0

Sean A. Irvine, Oct 14 2023

nonn

			a(3)=84 because 4^3+1 has divisors {1, 5, 13, 65}.

Max Alekseyev, Table of n, a(n) for n = 0..583

Cf. A000203, A052539, A057940, A274903, A366603, A366605, A366606, A366608, A366609.

Cf. A069061, A366578, A366617, A366629, A366638, A366657, A366666, A366668, A366689, A366715.

a:=n->numtheory[sigma](4^n+1):
seq(a(n), n=0..100);

DivisorSigma[1,4^Range[0,30]+1] (* Paolo Xausa, Oct 14 2023 *)

from sympy import divisor_sigma
def A366607(n): return divisor_sigma((1<<(n<<1))+1) # Chai Wah Wu, Oct 14 2023

a(n) = sigma(4^n+1) = A000203(A052539(n)).

a(n) = A069061(2*n). - Max Alekseyev, Jan 08 2024

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

A366605 Number of distinct prime divisors of 4^n + 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 3, 3, 4, 2, 5, 2, 4, 4, 4, 2, 6, 3, 5, 3, 5, 3, 6, 3, 3, 4, 5, 2, 6, 3, 6, 5, 5, 4, 9, 3, 5, 5, 5, 4, 10, 2, 4, 3, 6, 6, 9, 2, 4, 6, 6, 5, 8, 3, 7, 6, 6, 4, 10, 2, 9, 7, 6, 4, 8, 4, 6, 7, 5, 2, 12, 4, 9, 5, 4, 4, 10, 4, 6, 8, 10

Offset: 0

Sean A. Irvine, Oct 14 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..583

Cf. A001221, A052539, A057940, A274903, A366604, A366606, A366607, A366608, A366609.

Cf. A046799, A366580, A366615, A366627, A366636, A366655, A366664, A119704, A366686, A366712.

PrimeNu[4^Range[0,100]+1] (* Paolo Xausa, Oct 14 2023 *)

for(n = 0, 100, print1(omega(4^n + 1), ", "))

from sympy import primenu
def A366605(n): return primenu((1<<(n<<1))+1) # Chai Wah Wu, Oct 14 2023

a(n) = omega(4^n+1) = A001221(A052539(n)).

a(n) = A046799(2*n). - Max Alekseyev, Jan 08 2024

A366606 Number of divisors of 4^n+1.

Original entry on oeis.org

2, 2, 2, 4, 2, 6, 4, 8, 2, 16, 4, 8, 8, 16, 4, 48, 4, 16, 16, 16, 4, 64, 8, 32, 8, 64, 8, 64, 8, 8, 16, 32, 4, 64, 12, 96, 32, 32, 16, 768, 8, 32, 32, 32, 16, 1536, 4, 16, 8, 64, 64, 512, 4, 16, 64, 96, 32, 256, 8, 128, 64, 64, 16, 1024, 4, 768, 128, 64, 16

Offset: 0

Sean A. Irvine, Oct 14 2023

nonn

			a(3)=4 because 4^3+1 has divisors {1, 5, 13, 65}.

Max Alekseyev, Table of n, a(n) for n = 0..583

Cf. A000005, A052539, A057940, A274903, A366602, A366605, A366607, A366608, A366609.

Cf. A046798, A366577, A366616, A366628, A366637, A366656, A366665, A344897, A366688, A366714.

a:=n->numtheory[tau](4^n+1):
seq(a(n), n=0..100);

DivisorSigma[0,4^Range[0,100]+1] (* Paolo Xausa, Oct 14 2023 *)

PARI
```
a(n) = numdiv(4^n+1);
```

from sympy import divisor_count
def A366606(n): return divisor_count((1<<(n<<1))+1) # Chai Wah Wu, Oct 14 2023

a(n) = sigma0(4^n+1) = A000005(A052539(n)).

a(n) = A046798(2*n). - Max Alekseyev, Jan 08 2024

A366609 Smallest prime dividing 4^n + 1.

Original entry on oeis.org

2, 5, 17, 5, 257, 5, 17, 5, 65537, 5, 17, 5, 97, 5, 17, 5, 641, 5, 17, 5, 257, 5, 17, 5, 193, 5, 17, 5, 257, 5, 17, 5, 274177, 5, 17, 5, 97, 5, 17, 5, 65537, 5, 17, 5, 257, 5, 17, 5, 641, 5, 17, 5, 257, 5, 17, 5, 449, 5, 17, 5, 97, 5, 17, 5, 59649589127497217

Offset: 0

Sean A. Irvine, Oct 14 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..1983

Bisection of A002586.

Cf. A002586, A038371, A052539, A074476, A274903, A366605, A366606, A366607, A366608, A366670, A366671, A366719.

cp's OEIS Frontend

A052539 a(n) = 4^n + 1.

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A002587 Largest prime factor of 2^n + 1.

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A057940 Number of prime factors of 4^n + 1 (counted with multiplicity).

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A366608 a(n) = phi(4^n+1), where phi is Euler's totient function (A000010).

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A274905 Largest prime factor of 8^n + 1.

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Magma

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A366607 Sum of the divisors of 4^n+1.

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Maple

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Python

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A002592 Largest prime factor of 9^n + 1.

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