A074478 -id:A074478 - cp's OEIS frontend

A074476 Largest prime factor of 3^n + 1.

2, 2, 5, 7, 41, 61, 73, 547, 193, 37, 1181, 661, 6481, 398581, 16493, 271, 21523361, 1021, 530713, 101917, 42521761, 2269, 570461, 23535794707, 769, 22996651, 4795973261, 19927, 647753, 5385997, 47763361, 22434744889, 926510094425921

Offset: 0

Rick L. Shepherd, Aug 23 2002

nonn

Table of n, a(n) for n = 0..691
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A006530, A034472, A074477 (largest prime factor of 3^n - 1), A002587 (largest prime factor of 2^n + 1), A074478 (largest prime factor of 5^n + 1).

[Maximum(PrimeDivisors(3^n+1)): n in [0..40]]; // Vincenzo Librandi, Aug 23 2013

Table[FactorInteger[3^n + 1][[-1, 1]], {n, 0, 40}] (* Vincenzo Librandi, Aug 23 2013 *)

for(n=0,35, v=factor(3^n+1); print1(v[matsize(v)[1],1],","))

a(n) = A006530(A034472(n)). - Amiram Eldar, Feb 01 2020

Terms to a(100) in b-file from Vincenzo Librandi, Aug 23 2013
a(101)-a(658) in b-file from Amiram Eldar, Feb 01 2020
a(659)-a(691) in b-file from Max Alekseyev, Apr 25 2022, Jul 25 2023

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

A057939 Number of prime factors of 5^n + 1 (counted with multiplicity).

Original entry on oeis.org

2, 2, 4, 2, 3, 3, 4, 3, 6, 4, 5, 3, 4, 3, 7, 3, 4, 5, 5, 4, 10, 4, 4, 4, 5, 5, 10, 3, 4, 7, 5, 4, 9, 6, 7, 6, 5, 4, 8, 5, 6, 6, 6, 3, 10, 3, 5, 5, 7, 7, 10, 5, 5, 6, 7, 7, 9, 3, 6, 6, 6, 4, 16, 4, 8, 7, 3, 7, 9, 7, 5, 6, 5, 5, 13, 5, 9, 10, 6, 6, 14, 6, 5, 7, 9, 5, 9, 7, 5, 12, 8, 4, 10, 6, 9, 7, 7, 7, 12

Offset: 1

Patrick De Geest, Oct 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..471 (first 451 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), this sequence (b=5), A057940 (b=4), A057941 (b=3), A054992 (b=2).

Cf. A001222, A034474, A057956, A074478.

PrimeOmega[5^Range[100]+1] (* Harvey P. Dale, Nov 27 2013 *)

a(n) = A057956(2n) - A057956(n). - T. D. Noe, Jun 19 2003

a(n) = A001222(A034474(n)). - Amiram Eldar, Feb 01 2020

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

A227575 Largest prime factor of 7^n + 1.

Original entry on oeis.org

2, 2, 5, 43, 1201, 191, 181, 911, 169553, 117307, 4021, 10746341, 1201, 228511817, 13564461457, 6568801, 47072139617, 29078814248401, 13841169553, 4058036683, 810221830361, 309079, 83960385389, 3421093417510114543, 33232924804801, 79787519018560501

Offset: 0

Michel Marcus, Aug 22 2013

nonn

			7^12 + 1 = 2*73*193*409*1201, so a(12) = 1201.

Table of n, a(n) for n = 0..387
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A006530, A034491.

Cf. A002587, A074249, A074476, A074478.

[Maximum(PrimeDivisors(7^n+1)): n in [0..30]]; // Bruno Berselli, Aug 23 2013

Table[FactorInteger[7^n + 1][[-1, 1]], {n, 0, 30}] (* Bruno Berselli, Aug 23 2013 *)

a(n) = f = factor(7^n + 1); f[#f~, 1]; \\ Michel Marcus, Aug 22 2013

a(n) = A006530(A034491(n)). - Vincenzo Librandi, Jul 12 2016

Terms to a(100) in b-file from Vincenzo Librandi, Jul 12 2016
a(101)-a(372) in b-file from Amiram Eldar, Feb 02 2020
a(373)-a(387) in b-file from Max Alekseyev, Apr 25 2022, Aug 30 2023

A366618 a(n) = phi(5^n+1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 2, 12, 36, 312, 1040, 7200, 25088, 183808, 557928, 4396800, 15333120, 121680000, 406812744, 2817007200, 8558784000, 76264519680, 254230063200, 1710194342400, 6349120596480, 47334145996800, 127169887444992, 1088029470747648, 3889097389599864

Offset: 0

Sean A. Irvine, Oct 14 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..471

Cf. A000010, A034474, A057939, A074478, A295502, A366615, A366616, A366617.

Cf. A053285, A366579, A366608, A366630, A366639, A366658, A366667, A366669, A366690, A366716.

EulerPhi[5^Range[0,30]+1] (* Harvey P. Dale, Jun 07 2025 *)

PARI
```
{a(n) = eulerphi(5^n+1)}
```

A366615 Number of distinct prime divisors of 5^n + 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 3, 6, 3, 4, 5, 5, 4, 8, 4, 4, 4, 5, 4, 7, 3, 4, 7, 5, 4, 8, 6, 7, 6, 5, 4, 7, 5, 6, 6, 6, 3, 8, 3, 5, 5, 7, 7, 9, 5, 5, 6, 7, 7, 8, 3, 6, 6, 6, 4, 13, 4, 8, 7, 3, 7, 8, 7, 5, 6, 5, 5, 12, 5, 9, 9, 6, 6, 10, 6, 5, 7, 9

Offset: 0

Sean A. Irvine, Oct 14 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..471

Cf. A001221, A034474, A057939, A074478, A366611, A366616, A366617, A366618.

Cf. A046799, A366580, A366605, A366627, A366636, A366655, A366664, A119704, A366686, A366712.

Table[PrimeNu[5^n+1],{n,0,90}] (* Harvey P. Dale, Apr 06 2025 *)

for(n = 0, 100, print1(omega(5^n + 1), ", "))

a(n) = omega(5^n+1) = A001221(A034474(n)).

A366616 Number of divisors of 5^n+1.

Original entry on oeis.org

2, 4, 4, 12, 4, 8, 8, 16, 8, 32, 16, 32, 8, 16, 8, 96, 8, 16, 32, 32, 16, 576, 16, 16, 16, 32, 24, 320, 8, 16, 128, 32, 16, 384, 64, 128, 64, 32, 16, 192, 32, 64, 64, 64, 8, 512, 8, 32, 32, 128, 128, 768, 32, 32, 64, 128, 128, 384, 8, 64, 64, 64, 16, 24576, 16

Offset: 0

Sean A. Irvine, Oct 14 2023

nonn

			a(3)=12 because 5^3+1 has divisors {1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126}.

Max Alekseyev, Table of n, a(n) for n = 0..471

Cf. A000005, A034474, A057939, A074478, A366612, A366607, A366615, A366617, A366618.

Cf. A046798, A366577, A366606, A366628, A366637, A366656, A366665, A344897, A366688, A366714.

a:=n->numtheory[tau](5^n+1):
seq(a(n), n=0..100);

DivisorSigma[0, 5^Range[0, 70] + 1] (* Paolo Xausa, Apr 20 2025 *)

PARI
```
a(n) = numdiv(5^n+1);
```

a(n) = sigma0(5^n+1) = A000005(A034474(n)).

A366617 Sum of the divisors of 5^n+1.

Original entry on oeis.org

3, 12, 42, 312, 942, 6264, 25284, 162000, 620460, 4961280, 16161768, 103442688, 367381884, 2441936064, 9859525284, 76963663296, 228970112844, 1526377433328, 6339280635408, 38199227335200, 144103649734968, 1285221510144000, 3894650946433800, 24349131482713344

Offset: 0

Sean A. Irvine, Oct 14 2023

nonn

			a(3)=312 because 5^3+1 has divisors {1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126}.

Max Alekseyev, Table of n, a(n) for n = 0..471

Cf. A000203, A034474, A057939, A074478, A366613, A366615, A366616, A366618.

Cf. A069061, A366578, A366607, A366629, A366638, A366657, A366666, A366668, A366689, A366715.

a:=n->numtheory[sigma](5^n+1):
seq(a(n), n=0..100);

DivisorSigma[1, 5^Range[0, 30] + 1] (* Paolo Xausa, Jul 03 2024 *)

a(n) = sigma(5^n+1) = A000203(A034474(n)).

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

cp's OEIS Frontend

A074476 Largest prime factor of 3^n + 1.

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A274903 Largest prime factor of 4^n + 1.

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A057939 Number of prime factors of 5^n + 1 (counted with multiplicity).

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A074479 Largest prime factor of 5^n - 1.

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A227575 Largest prime factor of 7^n + 1.

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A366618 a(n) = phi(5^n+1), where phi is Euler's totient function (A000010).

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A366615 Number of distinct prime divisors of 5^n + 1.

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