A274905 -id:A274905 - cp's OEIS frontend

A057936 Number of prime factors of 8^n + 1 (counted with multiplicity).

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

Offset: 1

Patrick De Geest, Oct 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..502 (first 354 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n+1): A057934 (b=10), A057935 (b=9), this sequence (b=8), A057937 (b=7), A057938 (b=6), A057939 (b=5), A057940 (b=4), A057941 (b=3), A054992 (b=2).

Cf. A001222, A057953, A062395, A274905.

f:=func; [f(8^n + 1):n in [1..110]]; // Marius A. Burtea, Feb 02 2020

PrimeOmega[8^Range[100]+1] (* Harvey P. Dale, Dec 16 2014 *)

a(n) = A057953(2n) - A057953(n). - T. D. Noe, Jun 19 2003

a(n) = A001222(A062395(n)) = A054992(3*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

A366655 Number of distinct prime divisors of 8^n + 1.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 3, 3, 3, 5, 4, 4, 3, 6, 5, 3, 5, 6, 4, 4, 5, 6, 4, 5, 6, 9, 6, 5, 4, 10, 4, 3, 7, 9, 10, 6, 6, 8, 5, 6, 6, 10, 5, 7, 9, 8, 6, 7, 6, 12, 9, 5, 5, 10, 10, 8, 6, 8, 7, 8, 3, 9, 10, 4, 10, 12, 7, 8, 6, 14, 7, 8, 5, 10, 10, 8, 11, 16, 5, 7, 10

Offset: 0

Sean A. Irvine, Oct 15 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..502

Cf. A062395, A001221, A057936, A274905, A366651, A366632, A366656, A366657, A366658, A366671.

Cf. A046799, A366580, A366605, A366615, A366627, A366636, A366664, A119704, A366686, A366712.

for(n = 0, 100, print1(omega(8^n + 1), ", "))

a(n) = omega(8^n+1) = A001221(A062395(n)).

a(n) = A046799(3*n). - Max Alekseyev, Jan 09 2024

A366658 a(n) = phi(8^n+1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 6, 48, 324, 3840, 19800, 186624, 1365336, 16515072, 84768120, 760320000, 5632621632, 64258375680, 366369658200, 3105655160832, 20140520400000, 280012271910912, 1495522910085120, 12824556668190720, 95907982079387520, 1080582572777472000, 5688765822212629632

Offset: 0

Sean A. Irvine, Oct 15 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..502

Cf. A062395, A000010, A057936, A274905, A366654, A366655, A366656, A366657, A366671.

Cf. A053285, A366579, A366608, A366618, A366630, A366639, A366667, A366669, A366690, A366716.

EulerPhi[8^Range[0, 21] + 1] (* Paul F. Marrero Romero, Oct 17 2023 *)

PARI
```
{a(n) = eulerphi(8^n+1)}
```

from sympy import totient
def A366658(n): return totient((1<<3*n)+1) # Chai Wah Wu, Oct 15 2023

a(n) = A000010(A062395(n)). - Paul F. Marrero Romero, Nov 06 2023

a(n) = A053285(3*n). - Max Alekseyev, Jan 09 2024

A366656 Number of divisors of 8^n+1.

Original entry on oeis.org

2, 3, 4, 8, 4, 12, 16, 12, 8, 20, 48, 24, 16, 12, 64, 64, 8, 48, 64, 24, 16, 64, 64, 24, 32, 96, 768, 192, 32, 24, 1536, 24, 8, 256, 512, 1536, 64, 96, 256, 64, 64, 96, 1024, 48, 128, 1280, 256, 96, 128, 96, 8192, 1024, 32, 48, 1024, 2304, 256, 192, 256, 192

Offset: 0

Sean A. Irvine, Oct 15 2023

nonn

			a(4)=4 because 8^4+1 has divisors {1, 17, 241, 4097}.

Max Alekseyev, Table of n, a(n) for n = 0..502

Cf. A062395, A000005, A057936, A274905, A366653, A366638, A366655, A366657, A366658.

Cf. A046798, A366577, A366606, A366616, A366628, A366637, A366665, A344897, A366688, A366714.

a:=n->numtheory[tau](8^n+1):
seq(a(n), n=0..100);

DivisorSigma[0, 8^Range[0,59] + 1] (* Paul F. Marrero Romero, Nov 12 2023 *)

PARI
```
a(n) = numdiv(8^n+1);
```

a(n) = sigma0(8^n+1) = A000005(A062395(n)).

a(n) = A046798(3*n). - Max Alekseyev, Jan 09 2024

A366657 Sum of the divisors of 8^n+1.

Original entry on oeis.org

3, 13, 84, 800, 4356, 51792, 351120, 3100240, 17041416, 211053040, 1494039792, 12611914848, 73234343952, 794382536272, 5936210280000, 60037292774400, 282937726148616, 3264911394064320, 24128875076496960, 208532141890460960, 1225825603154905104

Offset: 0

Sean A. Irvine, Oct 15 2023

nonn

			a(4)=4356 because 8^4+1 has divisors {1, 17, 241, 4097}.

Max Alekseyev, Table of n, a(n) for n = 0..502

Cf. A062395, A000203, A057936, A274905, A366653, A366655, A366656, A366658.

Cf. A069061, A366578, A366607, A366617, A366629, A366638, A366666, A366668, A366689, A366715.

a:=n->numtheory[sigma](8^n+1):
seq(a(n), n=0..100);

DivisorSigma[1, 8^Range[0,20]+1] (* Paul F. Marrero Romero, Nov 19 2023 *)

a(n) = sigma(8^n+1) = A000203(A062395(n)).

a(n) = A069061(3*n). - Max Alekseyev, Jan 09 2024

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

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

A366671 Smallest prime dividing 8^n + 1.

Original entry on oeis.org

2, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 193, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 641, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 193, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 769, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5

Offset: 0

Sean A. Irvine, Oct 15 2023

nonn

a(n) = 3 if n is odd. a(n) = 5 if n == 2 (mod 4). - Robert Israel, Nov 20 2023

Max Alekseyev, Table of n, a(n) for n = 0..16383

Cf. A002586, A366609, A052536, A366655, A274905, A366670, A038371, A366719.

P1000:= mul(ithprime(i),i= 4..1000):
f:= proc(n) local t;
  if n::odd then return 3 elif n mod 4 = 2 then return 5 fi;
  t:= igcd(8^n+1,P1000);
  if t <> 1 then min(numtheory:-factorset(t)) else min(numtheory:-factorset(8^n+1)) fi
end proc:
map(f, [$0..100]); # Robert Israel, Nov 20 2023

Table[FactorInteger[8^n + 1][[1,1]], {n, 0, 78}] (* Paul F. Marrero Romero, Oct 20 2023 *)

from sympy import primefactors
def A366671(n): return min(primefactors((1<<3*n)+1)) # Chai Wah Wu, Oct 16 2023

a(n) = A020639(A062395(n)). - Paul F. Marrero Romero, Oct 20 2023

a(n) = A002586(3*n) for n >= 1. - Robert Israel, Nov 20 2023

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

A057936 Number of prime factors of 8^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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A366655 Number of distinct prime divisors of 8^n + 1.

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A366658 a(n) = phi(8^n+1), where phi is Euler's totient function (A000010).

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A366656 Number of divisors of 8^n+1.

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A366657 Sum of the divisors of 8^n+1.

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A274908 Largest prime factor of 8^n - 1.

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