A366681 -id:A366681

A366685

a(n) = phi(11^n-1), where phi is Euler's totient function (A000010).

+10
15

4, 32, 432, 3840, 64400, 373248, 7613424, 56217600, 765889344, 6913984000, 114117380608, 599824465920, 13796450740800, 98909341090560, 1356399209088000, 11341872916070400, 202178811399717504, 1171410130065973248, 24463636179365818512, 176391086415667200000

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

OFFSET

1,1

LINKS

Max Alekseyev, Table of n, a(n) for n = 1..316

MATHEMATICA

EulerPhi[11^Range[30] - 1]

PROG

(PARI) {a(n) = eulerphi(11^n-1)}

CROSSREFS

phi(k^n-1): A053287 (k=2), A295500 (k=3), A295501 (k=4), A295502 (k=5), A366623 (k=6), A366635 (k=7), A366654 (k=8), A366663 (k=9), A295503 (k=10), this sequence (k=11), A366711 (k=12).

Cf. A000010, A024127, A366681, A366682, A366683, A366684.

KEYWORD

nonn

AUTHOR

Sean A. Irvine, Oct 16 2023

STATUS

approved

A366683

Number of divisors of 11^n-1.

+10
14

4, 16, 16, 40, 12, 192, 16, 96, 32, 96, 16, 1920, 16, 128, 96, 448, 8, 1024, 8, 480, 768, 1024, 32, 18432, 128, 512, 64, 2560, 16, 9216, 32, 2048, 512, 256, 192, 20480, 64, 512, 4096, 4608, 512, 36864, 16, 10240, 384, 2048, 32, 1376256, 128, 4096, 512, 2560

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

OFFSET

1,1

LINKS

Max Alekseyev, Table of n, a(n) for n = 1..316

FORMULA

a(n) = sigma0(11^n-1) = A000005(A024127(n)).

EXAMPLE

a(3)=16 because 11^3-1 has divisors {1, 2, 5, 7, 10, 14, 19, 35, 38, 70, 95, 133, 190, 266, 665, 1330}.

MAPLE

a:=n->numtheory[tau](11^n-1):

seq(a(n), n=1..100);

MATHEMATICA

DivisorSigma[0, 11^Range[100]-1]

PROG

(PARI) a(n) = numdiv(11^n-1);

CROSSREFS

Cf. A024127, A000005, A366681, A366682, A366684, A366685.

Cf. A046801, A070528, A366709.

KEYWORD

nonn

AUTHOR

Sean A. Irvine, Oct 16 2023

STATUS

approved

A366684

Sum of the divisors of 11^n-1.

+10
14

18, 360, 2880, 46128, 299646, 7113600, 35893440, 686393568, 5105934720, 80436972240, 513593801496, 14266630210560, 62197735384584, 1165770116121600, 9349887314805120, 157025981601707904, 909804651298728804, 22898038082582016000, 110086362807146183340

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

OFFSET

1,1

LINKS

Max Alekseyev, Table of n, a(n) for n = 1..316

FORMULA

a(n) = sigma(11^n-1) = A000203(A024127(n)).

EXAMPLE

a(3)=2880 because 11^3-1 has divisors {1, 2, 5, 7, 10, 14, 19, 35, 38, 70, 95, 133, 190, 266, 665, 1330}.

MAPLE

a:=n->numtheory[sigma](11^n-1):

seq(a(n), n=1..100);

MATHEMATICA

DivisorSigma[1, 11^Range[30]-1]

CROSSREFS

Cf. A024127, A000203, A366710, A366666, A366681, A366682, A366683, A366685.

KEYWORD

nonn

AUTHOR

Sean A. Irvine, Oct 16 2023

STATUS

approved

A366707

Number of distinct prime divisors of 12^n - 1.

+10
14

1, 2, 2, 4, 2, 5, 3, 6, 4, 4, 3, 8, 3, 6, 6, 9, 3, 9, 2, 7, 5, 5, 4, 12, 4, 7, 6, 10, 5, 13, 5, 11, 7, 6, 9, 14, 3, 6, 7, 13, 4, 13, 5, 11, 12, 8, 3, 18, 5, 10, 6, 12, 7, 16, 7, 13, 7, 8, 4, 18, 4, 8, 8, 13, 8, 16, 5, 10, 7, 14, 4, 21, 3, 7, 11, 11, 10, 17, 4

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

OFFSET

1,2

LINKS

Max Alekseyev, Table of n, a(n) for n = 1..310

FORMULA

a(n) = omega(12^n-1) = A001221(A024140(n)).

PROG

(PARI) for(n = 1, 100, print1(omega(12^n - 1), ", "))

CROSSREFS

Cf. A024140, A001221, A046800, A133801, A102347, A250288, A252170, A366708, A366709, A366681.

KEYWORD

nonn

AUTHOR

Sean A. Irvine, Oct 17 2023

STATUS

approved

A366686

Number of distinct prime divisors of 11^n + 1.

+10
12

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

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

OFFSET

0,2

LINKS

Max Alekseyev, Table of n, a(n) for n = 0..325

FORMULA

a(n) = omega(11^n+1) = A001221(A034524(n)).

PROG

(PARI) for(n = 0, 100, print1(omega(11^n + 1), ", "))

CROSSREFS

Cf. A034524, A001221, A366681, A102347, A366687, A366688, A366689, A366690.

Cf. A046799, A119704, A366712.

KEYWORD

nonn

AUTHOR

Sean A. Irvine, Oct 16 2023

STATUS

approved

A366632

Number of distinct prime divisors of 7^n - 1.

+10
11

2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 4, 7, 3, 6, 6, 6, 4, 7, 4, 8, 6, 6, 5, 11, 5, 5, 9, 8, 5, 10, 5, 8, 8, 5, 7, 11, 5, 6, 7, 11, 5, 11, 4, 10, 10, 6, 4, 14, 8, 8, 9, 8, 5, 12, 6, 13, 8, 6, 6, 17, 6, 8, 9, 11, 9, 13, 6, 9, 9, 15, 4, 18, 7, 7, 10, 8, 9, 13, 4, 16, 13

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

OFFSET

1,1

LINKS

Max Alekseyev, Table of n, a(n) for n = 1..388

FORMULA

a(n) = omega(7^n-1) = A001221(A024075(n)).

PROG

(PARI) for(n = 1, 100, print1(omega(7^n - 1), ", "))

CROSSREFS

Cf. A024075, A001221, A057954, A059889, A074249, A218358, A366620, A366633, A366634, A366635.

Cf. A046800, A133801, A366604, A366611, A366620, A366651, A366660, A102347, A366681, A366707.

KEYWORD

nonn

AUTHOR

Sean A. Irvine, Oct 14 2023

STATUS

approved

A366660

Number of distinct prime divisors of 9^n - 1.

+10
11

1, 2, 3, 3, 3, 5, 3, 5, 6, 5, 5, 7, 3, 6, 8, 6, 6, 9, 5, 7, 8, 8, 4, 12, 7, 6, 11, 9, 7, 12, 6, 7, 10, 9, 8, 12, 6, 8, 12, 11, 6, 14, 4, 12, 16, 7, 8, 15, 10, 12, 13, 9, 6, 15, 11, 14, 13, 10, 5, 18, 5, 10, 16, 8, 9, 15, 6, 13, 13, 15, 7, 19, 7, 10, 19, 13, 11

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

OFFSET

1,2

LINKS

Max Alekseyev, Table of n, a(n) for n = 1..690

FORMULA

a(n) = omega(9^n-1) = A001221(A024101(n)).

a(n) = A133801(2*n) = A133801(n) + A366580(n) - 1. - Max Alekseyev, Jan 07 2024

PROG

(PARI) for(n = 1, 100, print1(omega(9^n - 1), ", "))

CROSSREFS

Cf. A024101, A001221, A057952, A366661, A366662, A366663.

Cf. A046800, A133801, A366604, A366611, A366620, A366632, A366651, A102347, A366681, A366707.

KEYWORD

nonn

AUTHOR

Sean A. Irvine, Oct 15 2023

STATUS

approved

A366604

Number of distinct prime divisors of 4^n - 1.

+10
10

1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 6, 3, 6, 6, 5, 3, 8, 3, 7, 6, 7, 4, 9, 7, 7, 6, 8, 6, 11, 3, 7, 8, 7, 9, 12, 5, 7, 7, 9, 5, 12, 5, 10, 11, 9, 6, 12, 5, 12, 10, 10, 6, 12, 11, 11, 8, 9, 6, 15, 3, 8, 11, 9, 9, 14, 5, 10, 8, 15, 6, 17, 6, 10, 13, 11, 10, 16, 5

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

OFFSET

1,2

LINKS

Max Alekseyev, Table of n, a(n) for n = 1..1122

FORMULA

a(n) = omega(4^n-1) = A001221(A024036(n)).

a(n) = A046800(2*n) = A046799(n) + A046800(n). - Max Alekseyev, Jan 07 2024

MATHEMATICA

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

PROG

(PARI) for(n = 1, 100, print1(omega(4^n - 1), ", "))

(Python)

from sympy import primenu

def A366604(n): return primenu((1<<(n<<1))-1) # Chai Wah Wu, Oct 15 2023

CROSSREFS

Cf. A024036, A001221, A057957, A133801, A366602, A366603.

Cf. A046800, A133801, A366611, A366620, A366632, A366651, A366660, A102347, A366681, A366707.

KEYWORD

nonn

AUTHOR

Sean A. Irvine, Oct 14 2023

STATUS

approved

A366620

Number of distinct prime divisors of 6^n - 1.

+10
10

1, 2, 2, 3, 2, 4, 2, 4, 4, 5, 3, 7, 3, 5, 5, 6, 5, 7, 3, 8, 4, 5, 5, 9, 4, 7, 6, 8, 2, 10, 3, 9, 6, 8, 6, 13, 6, 6, 6, 11, 3, 9, 5, 9, 10, 8, 4, 13, 5, 8, 9, 11, 4, 11, 6, 13, 7, 6, 4, 19, 4, 5, 10, 12, 8, 12, 3, 11, 8, 16, 2, 18, 5, 10, 10, 9, 6, 15, 4, 16, 8

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

OFFSET

1,2

LINKS

Max Alekseyev, Table of n, a(n) for n = 1..420

FORMULA

a(n) = omega(6^n-1) = A001221(A024062).

PROG

(PARI) for(n = 1, 100, print1(omega(6^n - 1), ", "))

CROSSREFS

Cf. A024062, A001221, A057955, A059888, A274907, A366621, A366622, A366623.

Cf. A046800, A133801, A366604, A366611, A366632, A366651, A366660, A102347, A366681, A366707.

KEYWORD

nonn

AUTHOR

Sean A. Irvine, Oct 14 2023

STATUS

approved

A366651

Number of distinct prime divisors of 8^n - 1.

+10
10

1, 2, 2, 4, 3, 4, 3, 6, 3, 6, 4, 8, 4, 6, 6, 9, 5, 6, 4, 11, 6, 8, 4, 12, 7, 7, 6, 12, 6, 11, 3, 12, 8, 10, 10, 12, 6, 8, 9, 15, 5, 11, 5, 14, 10, 8, 6, 17, 5, 13, 8, 16, 8, 12, 10, 17, 7, 10, 6, 21, 5, 7, 9, 15, 8, 15, 6, 19, 9, 20, 7, 18, 7, 12, 14, 16, 9

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

OFFSET

1,2

LINKS

Max Alekseyev, Table of n, a(n) for n = 1..500

FORMULA

a(n) = omega(8^n-1) = A001221(A024088(n)).

a(n) = A046800(3*n). - Max Alekseyev, Jan 09 2024

PROG

(PARI) for(n = 1, 100, print1(omega(8^n - 1), ", "))

CROSSREFS

Cf. A024088, A001221, A057953, A366652, A366653, A366654.

Cf. A046800, A133801, A366604, A366611, A366620, A366632, A366660, A102347, A366681, A366707.

KEYWORD

nonn

AUTHOR

Sean A. Irvine, Oct 15 2023

STATUS

approved