A366719 -id:A366719

A366714

Number of divisors of 12^n+1.

+10
14

2, 2, 4, 8, 4, 4, 8, 8, 8, 32, 12, 4, 16, 24, 16, 128, 4, 8, 32, 16, 64, 384, 64, 16, 64, 64, 32, 1024, 8, 8, 48, 8, 4, 512, 16, 32, 128, 16, 32, 1536, 16, 32, 64, 32, 16, 4096, 8, 32, 32, 32, 512, 512, 32, 32, 1024, 128, 512, 1536, 192, 64, 1024, 32, 64

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Max Alekseyev, Table of n, a(n) for n = 0..306

FORMULA

a(n) = sigma0(12^n+1) = A000005(A178248(n)).

EXAMPLE

a(4)=4 because 12^4+1 has divisors {1, 89, 233, 20737}.

MAPLE

a:=n->numtheory[tau](12^n+1):

seq(a(n), n=0..100);

PROG

(PARI) a(n) = numdiv(12^n+1);

CROSSREFS

Cf. A178248, A000005, A046798, A344897, A366709, A366712, A366713, A366715, A366716, A366719, A366720, A366688.

KEYWORD

nonn

AUTHOR

Sean A. Irvine, Oct 17 2023

STATUS

approved

A366716

a(n) = phi(12^n+1), where phi is Euler's totient function (A000010).

+10
14

1, 12, 112, 1296, 20416, 229680, 2306304, 32916240, 400515072, 3863116800, 47825825600, 685853880624, 8732596764672, 97509650382144, 990242755633152, 11148606564480000, 184883057981234176, 2047145911595946000, 20281543142263603200, 294779525244632305920

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Max Alekseyev, Table of n, a(n) for n = 0..306

FORMULA

a(n) = A000010(A178248(n)). - Paul F. Marrero Romero, Oct 27 2023

MATHEMATICA

EulerPhi[12^Range[0, 19] + 1] (* Paul F. Marrero Romero, Oct 27 2023 *)

PROG

(PARI) {a(n) = eulerphi(12^n+1)}

CROSSREFS

Cf. A178248, A000010, A366711, A366712, A366713, A366714, A366715, A366719, A366720, A366690.

KEYWORD

nonn

AUTHOR

Sean A. Irvine, Oct 17 2023

STATUS

approved

A366715

Sum of the divisors of 12^n+1.

+10
13

3, 14, 180, 2240, 21060, 267988, 3706920, 38773952, 459970056, 6692483840, 79425033660, 800162860756, 9101898907920, 117326869641600, 1596198064568400, 20655000929239040, 184885459808838660, 2390210102271311936, 33504016991491136160, 344201347103878781440

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Max Alekseyev, Table of n, a(n) for n = 0..306

FORMULA

a(n) = sigma(12^n+1) = A000203(A178248(n)).

EXAMPLE

a(4)=21060 because 12^4+1 has divisors {1, 89, 233, 20737}.

MAPLE

a:=n->numtheory[sigma](12^n+1):

seq(a(n), n=0..100);

CROSSREFS

Cf. A178248, A000203, A366710, A366712, A366713, A366714, A366716, A366719, A366720, A366689.

KEYWORD

nonn

AUTHOR

Sean A. Irvine, Oct 17 2023

STATUS

approved

A366609

Smallest prime dividing 4^n + 1.

+10
8

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

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

CROSSREFS

Bisection of A002586.

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

KEYWORD

nonn

AUTHOR

Sean A. Irvine, Oct 14 2023

STATUS

approved

A366713

Number of prime factors of 12^n + 1 (counted with multiplicity).

+10
8

1, 1, 2, 3, 2, 2, 3, 3, 3, 5, 4, 2, 4, 5, 4, 7, 2, 3, 5, 4, 6, 9, 6, 4, 6, 6, 5, 10, 3, 3, 6, 3, 2, 9, 4, 5, 7, 4, 5, 11, 4, 5, 6, 5, 4, 12, 3, 5, 5, 5, 10, 9, 5, 5, 10, 7, 9, 11, 8, 6, 10, 5, 6, 15, 5, 9, 11, 4, 5, 12, 10, 3, 10, 5, 8, 17, 5, 6, 9, 4, 6, 15

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Max Alekseyev, Table of n, a(n) for n = 0..306

FORMULA

a(n) = bigomega(12^n+1) = A001222(A178248(n)).

MATHEMATICA

PrimeOmega[12^Range[70]+1]

PROG

(PARI) a(n)=bigomega(12^n+1)

CROSSREFS

Cf. A178248, A001222, A054992, A057934, A057935, A057936, A057937, A057938, A057939, A057940, A057941, A366712, A366714, A366715, A366716, A366719, A366720, A366708, A366687.

KEYWORD

nonn

AUTHOR

Sean A. Irvine, Oct 17 2023

STATUS

approved

A366720

Largest prime factor of 12^n+1.

+10
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Max Alekseyev, Table of n, a(n) for n = 0..306

FORMULA

a(n) = A006530(A178248(n)). - Paul F. Marrero Romero, Dec 07 2023

MATHEMATICA

Table[FactorInteger[12^n + 1][[-1, 1]], {n, 0, 20}]

CROSSREFS

Cf. A178248, A006530, A002587, A074476, A274903, A074478, A274904, A227575, A274905, A002592, A003021, A062308, A002590, A366712, A366713, A366714, A366715, A366716, A366717, A366718, A366719, A324941.

KEYWORD

nonn

AUTHOR

Sean A. Irvine, Oct 17 2023

STATUS

approved

A366670

Smallest prime dividing 6^n + 1.

+10
6

2, 7, 37, 7, 1297, 7, 13, 7, 17, 7, 37, 7, 1297, 7, 37, 7, 353, 7, 13, 7, 41, 7, 37, 7, 17, 7, 37, 7, 281, 7, 13, 7, 2753, 7, 37, 7, 577, 7, 37, 7, 17, 7, 13, 7, 89, 7, 37, 7, 193, 7, 37, 7, 1297, 7, 13, 7, 17, 7, 37, 7, 41, 7, 37, 7, 4926056449, 7, 13, 7, 137

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Max Alekseyev, Table of n, a(n) for n = 0..4095

FORMULA

a(n) = A020639(A062394(n)). - Paul F. Marrero Romero, Oct 17 2023

MATHEMATICA

Table[FactorInteger[6^n + 1][[1, 1]], {n, 0, 68}] (* Paul F. Marrero Romero, Oct 17 2023 *)

CROSSREFS

Cf. A020639, A062394, A274904, A366630.

Cf. A002586, A366609, A366671, A038371, A366719.

KEYWORD

nonn

AUTHOR

Sean A. Irvine, Oct 15 2023

STATUS

approved

A366671

Smallest prime dividing 8^n + 1.

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

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

LINKS

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

FORMULA

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

MAPLE

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

MATHEMATICA

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

PROG

(Python)

from sympy import primefactors

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Sean A. Irvine, Oct 15 2023

STATUS

approved

A366717

Smallest prime dividing 12^n - 1.

+10
5

11, 11, 11, 5, 11, 7, 11, 5, 11, 11, 11, 5, 11, 11, 11, 5, 11, 7, 11, 5, 11, 11, 11, 5, 11, 11, 11, 5, 11, 7, 11, 5, 11, 11, 11, 5, 11, 11, 11, 5, 11, 7, 11, 5, 11, 11, 11, 5, 11, 11, 11, 5, 11, 7, 11, 5, 11, 11, 11, 5, 11, 11, 11, 5, 11, 7, 11, 5, 11, 11, 11

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Periodic with period 12, repeat of 11, 11, 11, 5, 11, 7, 11, 5, 11, 11, 11, 5.

LINKS

Table of n, a(n) for n=1..71.

FORMULA

a(n) = A020639(A024140(n)). - Paul F. Marrero Romero, Oct 25 2023

MATHEMATICA

Table[FactorInteger[12^n - 1][[1, 1]], {n, 71}] (* Paul F. Marrero Romero, Oct 25 2023 *)

CROSSREFS

Cf. A024140, A366718, A366719.

KEYWORD

nonn

AUTHOR

Sean A. Irvine, Oct 17 2023

STATUS

approved