A024127 -id:A024127 - cp's OEIS frontend

A366685 a(n) = phi(11^n-1), where phi is Euler's totient function (A000010).

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

Offset: 1

Sean A. Irvine, Oct 16 2023

nonn

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

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.

Mathematica
```
EulerPhi[11^Range[30] - 1]
```
PARI
```
{a(n) = eulerphi(11^n-1)}
```

A366683 Number of divisors of 11^n-1.

Original entry on oeis.org

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

Offset: 1

Sean A. Irvine, Oct 16 2023

nonn

			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}.

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

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

Cf. A046801, A070528, A366709.

a:=n->numtheory[tau](11^n-1):
seq(a(n), n=1..100);

Mathematica
```
DivisorSigma[0, 11^Range[100]-1]
```
PARI
```
a(n) = numdiv(11^n-1);
```

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

A366684 Sum of the divisors of 11^n-1.

Original entry on oeis.org

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

Offset: 1

Sean A. Irvine, Oct 16 2023

nonn

			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}.

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

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

a:=n->numtheory[sigma](11^n-1):
seq(a(n), n=1..100);

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

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

A366681 Number of distinct prime divisors of 11^n - 1.

Original entry on oeis.org

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

Offset: 1

Sean A. Irvine, Oct 16 2023

nonn

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

Cf. A024127, A001221, A274910, A366682, A366683, A366684, A366685.

Cf. A046800, A133801, A102347, A366707.

for(n = 1, 100, print1(omega(11^n - 1), ", "))

a(n) = omega(11^n-1) = A001221(A024127(n)).

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

Original entry on oeis.org

0, 11, 143, 1727, 20735, 248831, 2985983, 35831807, 429981695, 5159780351, 61917364223, 743008370687, 8916100448255, 106993205379071, 1283918464548863, 15407021574586367, 184884258895036415

Offset: 0

N. J. A. Sloane

In base 12 these are 0, B, BB, BBB, ... . - David Rabahy, Dec 12 2016

Index entries for linear recurrences with constant coefficients, signature (13,-12).

Cf. A001021, A016125, A178248.

Cf. Similar sequences of the type k^n-1: A000004 (k=1), A000225 (k=2), A024023 (k=3), A024036 (k=4), A024049 (k=5), A024062 (k=6), A024075 (k=7), A024088 (k=8), A024101 (k=9), A002283 (k=10), A024127 (k=11), this sequence (k=12).

12^Range[0,20]-1 (* or *) LinearRecurrence[{13,-12},{0,11},20] (* Harvey P. Dale, Feb 01 2019 *)

From Mohammad K. Azarian, Jan 14 2009: (Start)
G.f.: 1/(1-12*x) - 1/(1-x).
E.g.f.: exp(12*x) - exp(x). (End)
a(n) = 12*a(n-1) + 11 for n>0, a(0)=0. - Vincenzo Librandi, Nov 18 2010
a(n) = Sum_{i=1..n} 11^i*binomial(n,n-i) for n>0, a(0)=0. - Bruno Berselli, Nov 11 2015
From Elmo R. Oliveira, Dec 15 2023: (Start)
a(n) = 13*a(n-1) - 12*a(n-2) for n>1.
a(n) = A001021(n)-1 = A178248(n)-2.
a(n) = 11*(A016125(n) - 1)/12. (End)

A366682 Number of prime factors of 11^n - 1 (counted with multiplicity).

Original entry on oeis.org

2, 5, 4, 7, 4, 9, 4, 9, 5, 8, 4, 13, 4, 8, 7, 12, 3, 12, 3, 11, 10, 11, 5, 17, 8, 10, 6, 13, 4, 15, 5, 15, 9, 9, 8, 17, 6, 10, 12, 15, 9, 17, 4, 15, 9, 12, 5, 24, 7, 14, 9, 13, 6, 16, 10, 19, 8, 10, 5, 21, 5, 12, 16, 19, 8, 22, 6, 15, 10, 19, 7, 24, 3, 11, 15

Offset: 1

Sean A. Irvine, Oct 16 2023

nonn

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

Cf. A024127, A001222, A366681, A366683, A366684, A366685, A366687.

Cf. A046051, A057958, A057957, A057956, A057955, A057954, A057953, A057952, A057951, A366708.

Mathematica
```
PrimeOmega[11^Range[70]-1]
```
PARI
```
a(n)=bigomega(11^n-1)
```

a(n) = bigomega(11^n-1) = A001222(A024127(n)).

A274910 Largest prime factor of 11^n - 1.

Original entry on oeis.org

5, 5, 19, 61, 3221, 37, 45319, 7321, 1772893, 13421, 1806113, 1117, 3158528101, 1623931, 195019441, 6304673, 50544702849929377, 1772893, 6115909044841454629, 212601841, 45319, 1806113, 3740221981231, 20113, 1856458657451, 3158528101, 5559917315850179173

Offset: 1

Vincenzo Librandi, Jul 11 2016

nonn

			11^4 - 1 = 14640 = 2^4*3*5*61, so a(4) = 61.

Table of n, a(n) for n = 1..316
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

Cf. A006530, A024127.

Cf. similar sequences listed in A274906.

[Maximum(PrimeDivisors(11^n-1)): n in [1..40]];

Table[FactorInteger[11^n - 1][[-1, 1]], {n, 40}]

a(n) = A006530(A024127(n)).

Terms to a(70) in b-file from Vincenzo Librandi, Jul 13 2016
a(71)-a(306) in b-file from Amiram Eldar, Feb 08 2020
a(307)-a(316) in b-file from Max Alekseyev, Apr 25 2022, Oct 26 2023

A218336 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(11) listed in ascending order.

Original entry on oeis.org

1, 2, 5, 10, 3, 4, 6, 8, 12, 15, 20, 24, 30, 40, 60, 120, 7, 14, 19, 35, 38, 70, 95, 133, 190, 266, 665, 1330, 16, 48, 61, 80, 122, 183, 240, 244, 305, 366, 488, 610, 732, 915, 976, 1220, 1464, 1830, 2440, 2928, 3660, 4880, 7320, 14640, 25, 50, 3221, 6442

Offset: 1

Alois P. Heinz, Oct 26 2012

			Triangle begins:
   1,  2,    5,   10;
   3,  4,    6,    8,    12,    15,    20,     24,  30,  40, ...
   7, 14,   19,   35,    38,    70,    95,    133, 190, 266, ...
  16, 48,   61,   80,   122,   183,   240,    244, 305, 366, ...
  25, 50, 3221, 6442, 16105, 32210, 80525, 161050;
  ...

Alois P. Heinz, Rows n = 1..23, flattened
Eric Weisstein's World of Mathematics, Irreducible Polynomial
Eric Weisstein's World of Mathematics, Polynomial Order

Column k=5 of A212737.
Last elements of rows give: A024127.
Column k=1 gives: A218359.
Row lengths are A212957(n,11).

with(numtheory):
M:= proc(n) M(n):= divisors(11^n-1) minus U(n-1) end:
U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
T:= n-> sort([M(n)[]])[]:
seq(T(n), n=1..5);

M[n_] := M[n] = Divisors[11^n - 1] ~Complement~ U[n-1];
U[n_] := U[n] = If[n == 0, {}, M[n] ~Union~ U[n-1]];
T[n_] := Sort[M[n]];
Table[T[n], {n, 1, 5}] // Flatten (* Jean-François Alcover, Feb 12 2023, after Alois P. Heinz *)

T(n,k) = k-th smallest element of M(n) = {d : d|(11^n-1)} \ U(n-1) with U(n) = M(n) union U(n-1) if n>0, U(0) = {}.

A130652 a(n) = 11^n - 2.

Original entry on oeis.org

9, 119, 1329, 14639, 161049, 1771559, 19487169, 214358879, 2357947689, 25937424599, 285311670609, 3138428376719, 34522712143929, 379749833583239, 4177248169415649, 45949729863572159, 505447028499293769, 5559917313492231479, 61159090448414546289, 672749994932560009199

Offset: 1

Alexander Adamchuk, Jun 20 2007

There are only two known primes in a(n): a(4) = 14639 and a(6) = 1771559 (see A128472 = smallest prime of the form (2n-1)^k - 2 for k > (2n-1), or 0 if no such number exists). 3 divides a(2k-1). 7 divides a(3k-1). 13 divides a(12k-5). 17 divides a(16k-14).
Final digit of a(n) is 9.
Final two digits of a(n) are periodic with period 10: a(n) mod 100 = {09, 19, 29, 39, 49, 59, 69, 79, 89, 99}.
Final three digits of a(n) are periodic with period 50: a(n) mod 1000 = {009, 119, 329, 639, 049, 559, 169, 879, 689, 599, 609, 719, 929, 239, 649, 159, 769, 479, 289, 199, 209, 319, 529, 839, 249, 759, 369, 079, 889, 799, 809, 919, 129, 439, 849, 359, 969, 679, 489, 399, 409, 519, 729, 039, 449, 959, 569, 279, 089, 999}.

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Index entries for linear recurrences with constant coefficients, signature (12,-11).

Cf. A001020, A024127, A034524. Cf. A104096 = Largest prime <= 11^n. Cf. A084714 = smallest prime of the form (2n-1)^k - 2, or 0 if no such number exists. Cf. A128472 = smallest prime of the form (2n-1)^k - 2 for k>(2n-1), or 0 if no such number exists. Cf. A014224, A109080, A090669, A128455, A128457, A128458, A128459, A128460, A128461.

[11^n - 2: n in [1..50]]; // Vincenzo Librandi, Jun 08 2011

LinearRecurrence[{12, -11},{9, 119},17] (* Ray Chandler, Aug 26 2015 *)

a(n) = 11*a(n-1) + 20; a(1)=9. - Vincenzo Librandi, Jun 08 2011
From Elmo R. Oliveira, Jun 16 2025: (Start)
G.f.: x*(11*x+9)/((11*x-1)*(x-1)).
E.g.f.: 1 + exp(x)*(exp(10*x) - 2).
a(n) = 12*a(n-1) - 11*a(n-2) for n > 2. (End)

A249435 a(1) = 0, after which one less than prime powers p^m with exponent m >= 2.

Original entry on oeis.org

0, 3, 7, 8, 15, 24, 26, 31, 48, 63, 80, 120, 124, 127, 168, 242, 255, 288, 342, 360, 511, 528, 624, 728, 840, 960, 1023, 1330, 1368, 1680, 1848, 2047, 2186, 2196, 2208, 2400, 2808, 3124, 3480, 3720, 4095, 4488, 4912, 5040, 5328, 6240, 6560, 6858, 6888, 7920, 8191, 9408, 10200, 10608, 11448

Offset: 1

Antti Karttunen, Nov 02 2014

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

One less than A025475.
Subsequence of A181062 and also a subsequence of A249433 (after the initial zero).
Union of sequences A000225, A024023, A024049, A024075, A024127, etc. without their term a(1).
Apart from the first term, subsequence of A045542.

list(lim)=my(v=List([0])); lim=lim\1+1; for(m=2,logint(lim,2), forprime(p=2,sqrtnint(lim,m), listput(v, p^m-1))); Set(v) \\ Charles R Greathouse IV, Aug 26 2015

Scheme
```
(define (A249435 n) (- (A025475 n) 1))
```

a(n) = A025475(n) - 1.

cp's OEIS Frontend

A366685 a(n) = phi(11^n-1), where phi is Euler's totient function (A000010).

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PARI

A366683 Number of divisors of 11^n-1.

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Maple

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A366684 Sum of the divisors of 11^n-1.

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Maple

Mathematica

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A366681 Number of distinct prime divisors of 11^n - 1.

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PARI

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

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Mathematica

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A366682 Number of prime factors of 11^n - 1 (counted with multiplicity).

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Mathematica

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Formula

A274910 Largest prime factor of 11^n - 1.

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Magma

Mathematica

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Extensions

A218336 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(11) listed in ascending order.

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