A053287 -id:A053287 - cp's OEIS frontend

A058891 a(n) = 2^(2^(n-1) - 1).

1, 2, 8, 128, 32768, 2147483648, 9223372036854775808, 170141183460469231731687303715884105728, 57896044618658097711785492504343953926634992332820282019728792003956564819968

Offset: 1

N. J. A. Sloane, Jan 08 2001

For n > 1, a(n) is the least solution > 1 to rad(x)^(n-1) = tau(x) where rad(x) = A007947(x) is the squarefree kernel of x and tau(x) = A000005(x) the number of divisors of x. - Benoit Cloitre, Apr 18 2002 [Corrected by Michel Marcus, Oct 15 2018]
For n > 1, a(n) is also the total number of possible outcomes of a knockout tournament starting with 2^(n-1) players, taking account of all matches in the tournament. - Martin Griffiths, Mar 26 2009
Also, a(n+1) = 2^(2^n-1) for n >= 1 are solutions x = y of the Diophantine equation x^y * y^x = (x+y)^z in positive integers; corresponding solutions z are in A348332 (see this last sequence for more informations and links). - Bernard Schott, Oct 13 2021
For n > 2, a(n) ends with 8. - Bernard Schott, Oct 20 2021
a(n) is the number of labeled hypergraphs on n - 1 vertices. - Lorenzo Sauras Altuzarra, Apr 01 2023

			The 8 possible hyperedge sets for the vertex set {1, 2} are {}, {{1}}, {{2}}, {{1, 2}}, {{1}, {2}}, {{1}, {1, 2}}, {{2}, {1, 2}} and {{1}, {2}, {1, 2}}. - _Lorenzo Sauras Altuzarra_, Apr 01 2023

F. Harary, Graph Theory, Page 209, Problem 16.11.

Harry J. Smith, Table of n, a(n) for n = 1..12
Wikipedia, Hypergraph.

Cf. A000079, A000225, A053287, A076214, A077585, A348332.

a[1]:=1: for n from 2 to 20 do a[n]:=2*a[n-1]^2 od: seq(a[n], n=1..9); # Zerinvary Lajos, Apr 16 2009

a = 1; b = -3; Table[Expand[(-1/2) ((a + Sqrt[b])^(2^n) + (a - Sqrt[b])^(2^n))], {n, 1, 10}] (* Artur Jasinski, Oct 11 2008 *)

a(n) = { 2^(2^(n-1)-1) } \\ Harry J. Smith, Jun 23 2009

def A058891(n): return 1<<(1<Chai Wah Wu, Dec 12 2022

a(n) = A053287(A000079(n-1)).
a(1) = 1, a(n+1) = 2*a(n)^2.
a(1) = 1, a(n+1) = 2^n*a(1)*a(2)*...*a(n). - Benoit Cloitre, Sep 13 2003
a(n) = (-1/2)*((1 + sqrt(-3))^(2^n) + (1 - sqrt(-3))^(2^n)). - Artur Jasinski, Oct 11 2008
a(n) = 2*a(n-1)^2 is an example with a(1) = 1 and k = 2 of a(n) = k*a(n-1)^2; general explicit formula: a(n) = ((a(1)*k)^(2^(n-1)))/k. - Andreas Pfaffel (andreas.pfaffel(AT)gmx.at), Apr 27 2010
a(n) = A077585(n-1) + 1. - Maurizio De Leo, Feb 25 2015
a(n) = 2^A000225(n-1). - Michel Marcus, Aug 19 2020
Sum_{n>=0} 1/a(n) = A076214. - Amiram Eldar, Oct 27 2020

A295501 a(n) = phi(4^n-1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

2, 8, 36, 128, 600, 1728, 10584, 32768, 139968, 480000, 2640704, 6635520, 44717400, 132765696, 534600000, 2147483648, 11452896600, 26121388032, 183250539864, 473702400000, 2427720325632, 8834232287232, 45914084232320, 109586090557440, 656100000000000

Offset: 1

Seiichi Manyama, Nov 22 2017

nonn

Max Alekseyev, Table of n, a(n) for n = 1..1128
Eric Weisstein's World of Mathematics, Totient Function.
Wikipedia, Euler's totient function.

Cf. A000010, A053285.
phi(k^n-1): A053287 (k=2), A295500 (k=3), this sequence (k=4), A295502 (k=5), A366623 (k=6), A366635 (k=7), A366654 (k=8), A366663 (k=9), A295503 (k=10), A366685 (k=11), A366711 (k=12).
Cf. A366602, A366603, A366604, A366608.

EulerPhi[4^Range[30] - 1] (* Paolo Xausa, Jun 17 2024 *)

PARI
```
{a(n) = eulerphi(4^n-1)}
```

a(n) = n*A027695(n).

a(n) = A053287(2*n) = A053285(n) * A053287(n). - Max Alekseyev, Jan 07 2024

A366623 a(n) = phi(6^n-1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

4, 24, 168, 864, 6200, 30240, 223944, 1119744, 7457184, 37200000, 277618528, 1254113280, 10445497920, 51618196224, 365601600000, 1770091315200, 13439285266176, 62336092492800, 484935499902880, 2179146240000000, 17141125020596640, 86330728271779200

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..430

phi(k^n-1): A053287 (k=2), A295500 (k=3), A295501 (k=4), A295502 (k=5), this sequence (k=6), A366635 (k=7), A366654 (k=8), A366663 (k=9), A295503 (k=10), A366685 (k=11), A366711 (k=12).

Cf. A024062, A000010, A295502, A366620, A366621, A366622.

EulerPhi[6^Range[22] - 1] (* Paul F. Marrero Romero, Oct 23 2023 *)

PARI
```
{a(n) = eulerphi(6^n-1)}
```

a(n) = A000010(A024062(n)). - Paul F. Marrero Romero, Oct 23 2023

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

Original entry on oeis.org

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)}
```

A295502 a(n) = phi(5^n-1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

2, 8, 60, 192, 1400, 4320, 39060, 119808, 894240, 2912000, 24414060, 62208000, 610351560, 1959874560, 13154400000, 44043337728, 380537036928, 997843069440, 9485297382000, 25606963200000, 230106651919200, 748687423334400, 5959800062798400, 15138938880000000

Offset: 1

Seiichi Manyama, Nov 22 2017

nonn

Faye et al. prove that no term is of the form 5^k-1. - Michel Marcus, Jun 16 2024

Max Alekseyev, Table of n, a(n) for n = 1..502
Bernadette Faye, Florian Luca, and Amadou Tall, On the equation phi(5^m-1)=5^n-1, Bull. Korean Math. Soc. 2015; 52(2): 513-524.
Eric Weisstein's World of Mathematics, Totient Function
Wikipedia, Euler's totient function

Cf. A000010, A024049.

phi(k^n-1): A053287 (k=2), A295500 (k=3), A295501 (k=4), this sequence (k=5), A366623 (k=6), A366635 (k=7), A366654 (k=8), A366663 (k=9), A295503 (k=10), A366685 (k=11), A366711 (k=12).

EulerPhi[5^Range[25] - 1] (* Paolo Xausa, Jun 18 2024 *)

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

a(n) = n*A027741(n).

a(n) = A000010(A024049(n)). - Michel Marcus, Jun 16 2024

A366635 a(n) = phi(7^n-1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

2, 16, 108, 640, 5600, 36288, 264992, 1536000, 12387168, 85120000, 658519752, 3135283200, 32296336800, 216063877120, 1450340640000, 8333819904000, 77537969371008, 488237947481088, 3790563394976072, 19162214400000000, 170264753751665664, 1245495178700551680

Offset: 1

Sean A. Irvine, Oct 15 2023

nonn

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

phi(k^n-1): A053287 (k=2), A295500 (k=3), A295501 (k=4), A295502 (k=5), A366623 (k=6), this sequence (k=7), A366654 (k=8), A366663 (k=9), A295503 (k=10), A366685 (k=11), A366711 (k=12).

Cf. A000010, A024075, A366632, A366633, A366634.

EulerPhi[7^Range[30] - 1] (* Wesley Ivan Hurt, Oct 15 2023 *)

PARI
```
{a(n) = eulerphi(7^n-1)}
```

A366654 a(n) = phi(8^n-1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

6, 36, 432, 1728, 27000, 139968, 1778112, 6635520, 113467392, 534600000, 6963536448, 26121388032, 465193834560, 2427720325632, 28548223200000, 109586090557440, 1910296842179040, 9618417501143040, 123523151337020736, 406467072000000000, 7713001620195508224

Offset: 1

Sean A. Irvine, Oct 15 2023

nonn

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

phi(k^n-1): A053287 (k=2), A295500 (k=3), A295501 (k=4), A295502 (k=5), A366623 (k=6), A366635 (k=7), this sequence (k=8), A366663 (k=9), A295503 (k=10), A366685 (k=11), A366711 (k=12).

Cf. A000010, A010705, A024088, A057953, A059890, A274908, A366651, A366652, A366653.

Mathematica
```
EulerPhi[8^Range[30] - 1]
```
PARI
```
{a(n) = eulerphi(8^n-1)}
```

from sympy import totient
def A366654(n): return totient((1<<3*n)-1) # Chai Wah Wu, Oct 15 2023

a(n) = A053287(3*n). - Max Alekseyev, Jan 09 2024

A366663 a(n) = phi(9^n-1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

4, 32, 288, 2560, 26400, 165888, 2384928, 15728640, 141087744, 1246080000, 14758128000, 85996339200, 1270928131200, 8810420097024, 70207948800000, 677066362060800, 8218041445152000, 43129128265187328, 674757689572915200, 4238841176064000000

Offset: 1

Sean A. Irvine, Oct 15 2023

nonn

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

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), this sequence (k=9), A295503 (k=10), A366685 (k=11), A366711 (k=12).

Cf. A000010, A024101, A057952, A366660, A366661, A366662, A366667.

Mathematica
```
EulerPhi[9^Range[30] - 1]
```
PARI
```
{a(n) = eulerphi(9^n-1)}
```

a(n) = A295500(2*n) = 2 * A295500(n) * A366579(n). - Max Alekseyev, Jan 07 2024

A295500 a(n) = phi(3^n-1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 4, 12, 32, 110, 288, 1092, 2560, 9072, 26400, 84700, 165888, 797160, 2384928, 6019200, 15728640, 64533700, 141087744, 580765248, 1246080000, 4823425152, 14758128000, 46070066188, 85996339200, 385087175000, 1270928131200, 3474144608256, 8810420097024

Offset: 1

Seiichi Manyama, Nov 22 2017

nonn

Max Alekseyev, Table of n, a(n) for n = 1..690
Eric Weisstein's World of Mathematics, Totient Function
Wikipedia, Euler's totient function

Cf. A000010, A024023, A027385.

phi(k^n-1): A053287 (k=2), this sequence (k=3), A295501 (k=4), A295502 (k=5), A366623 (k=6), A366635 (k=7), A366654 (k=8), A366663 (k=9), A295503 (k=10), A366685 (k=11), A366711 (k=12).

EulerPhi[3^Range[30] - 1] (* Paolo Xausa, Jun 18 2024 *)

PARI
```
{a(n) = eulerphi(3^n-1)}
```

a(n) = n*A027385(n).

a(n) = A000010(A024023(n)). - Michel Marcus, Jun 18 2024

A366711 a(n) = phi(12^n-1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

10, 120, 1560, 13440, 226200, 2021760, 32518360, 274391040, 4534807680, 51953616000, 646094232960, 4662793175040, 97266341877120, 1070382142166400, 13666309113600000, 109897747141754880, 2016918439151095000, 17518491733377024000, 290436363064202660760

Offset: 1

Sean A. Irvine, Oct 17 2023

nonn

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

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), A366685 (k=11), this sequence (k=12).

Cf. A000010, A024140, A366707, A366708, A366709, A366710, A366717, A366718.

Mathematica
```
EulerPhi[12^Range[30] - 1]
```
PARI
```
{a(n) = eulerphi(12^n-1)}
```

cp's OEIS Frontend

A058891 a(n) = 2^(2^(n-1) - 1).

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A295501 a(n) = phi(4^n-1), where phi is Euler's totient function (A000010).

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A366623 a(n) = phi(6^n-1), where phi is Euler's totient function (A000010).

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

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Mathematica

PARI

A295502 a(n) = phi(5^n-1), where phi is Euler's totient function (A000010).

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Programs

Mathematica

PARI

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A366635 a(n) = phi(7^n-1), where phi is Euler's totient function (A000010).

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Mathematica

PARI

A366654 a(n) = phi(8^n-1), where phi is Euler's totient function (A000010).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A366663 a(n) = phi(9^n-1), where phi is Euler's totient function (A000010).

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