A369291 -id:A369291 - cp's OEIS frontend

A027741 Number of primitive polynomials of degree n over GF(5).

1, 2, 4, 20, 48, 280, 720, 5580, 14976, 99360, 291200, 2219460, 5184000, 46950120, 139991040, 876960000, 2752708608, 22384531584, 55435726080, 499226178000, 1280348160000, 10957459615200, 34031246515200, 259121741860800, 630789120000000, 5255180000000000

Offset: 0

N. J. A. Sloane

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..100

Column k=5 of A369291.

with(numtheory): seq(`if`(n=0, 1, phi(5^n-1)/n), n=0..25);

Join[{1}, Array[EulerPhi[5^# - 1]/# &, 25]] (* Paolo Xausa, Jun 17 2024 *)

a(n) = if(n==0, 1, eulerphi(5^n-1)/n) \\ Andrew Howroyd, Feb 01 2024

a(23) onwards from Andrew Howroyd, Feb 01 2024

A027695 Number of primitive polynomials of degree n over GF(4).

Original entry on oeis.org

1, 2, 4, 12, 32, 120, 288, 1512, 4096, 15552, 48000, 240064, 552960, 3439800, 9483264, 35640000, 134217728, 673699800, 1451188224, 9644765256, 23685120000, 115605729792, 401556013056, 1996264531840, 4566087106560, 26244000000000, 89961392102400, 356237685227520

Offset: 0

N. J. A. Sloane

nonn

Amiram Eldar, Table of n, a(n) for n = 0..1122 (terms 0..150 from Andrew Howroyd)

Column k=4 of A369291.

Cf. A027742, A295501.

with(numtheory): seq(`if`(n=0, 1, phi(4^n-1)/n), n=0..27);

Join[{1}, Array[EulerPhi[4^# - 1]/# &, 30]] (* Paolo Xausa, Jun 17 2024 *)

a(n) = if(n==0, 1, eulerphi(4^n-1)/n) \\ Andrew Howroyd, Feb 01 2024

a(n) = A295501(n)/n = 2*A027742(n) for n >= 1. - Amiram Eldar, Nov 30 2024

a(24) onwards from Andrew Howroyd, Feb 01 2024

A027743 Number of primitive polynomials of degree n over GF(7).

Original entry on oeis.org

1, 2, 8, 36, 160, 1120, 6048, 37856, 192000, 1376352, 8512000, 59865432, 261273600, 2484333600, 15433134080, 96689376000, 520863744000, 4561057021824, 27124330415616, 199503336577688, 958110720000000, 8107845416745984, 56613417213661440, 388083262835207968

Offset: 0

N. J. A. Sloane

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..100

Column k=7 of A369291.

with(numtheory): seq(`if`(n=0, 1, phi(7^n-1)/n), n=0..23);

Join[{1}, Array[EulerPhi[7^# - 1]/# &, 25]] (* Paolo Xausa, Jun 17 2024 *)

a(n) = if(n==0, 1, eulerphi(7^n-1)/n) \\ Andrew Howroyd, Feb 01 2024

a(21) onwards from Andrew Howroyd, Feb 01 2024

A027744 Number of primitive polynomials of degree n over GF(8).

Original entry on oeis.org

1, 6, 18, 144, 432, 5400, 23328, 254016, 829440, 12607488, 53460000, 633048768, 2176782336, 35784141120, 173408594688, 1903214880000, 6849130659840, 112370402481120, 534356527841280, 6501218491422144, 20323353600000000, 367285791437881344, 1782862092373874688

Offset: 0

N. J. A. Sloane

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..100

Column k=8 of A369291.

with(numtheory): seq(`if`(n=0, 1, phi(8^n-1)/n), n=0..22);

Join[{1}, Array[EulerPhi[8^# - 1]/# &, 25]] (* Paolo Xausa, Jun 17 2024 *)

a(n) = if(n==0, 1, eulerphi(8^n-1)/n) \\ Andrew Howroyd, Feb 01 2024

a(20) onwards from Andrew Howroyd, Feb 01 2024

A027745 Number of primitive polynomials of degree n over GF(9).

Original entry on oeis.org

1, 4, 16, 96, 640, 5280, 27648, 340704, 1966080, 15676416, 124608000, 1341648000, 7166361600, 97763702400, 629315721216, 4680529920000, 42316647628800, 483414202656000, 2396062681399296, 35513562609100800, 211942058803200000, 2006922093666287616, 16837843397760000000

Offset: 0

N. J. A. Sloane

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..100

Column k=9 of A369291.

with(numtheory): seq(`if`(n=0, 1, phi(9^n-1)/n), n=0..22);

Join[{1}, Array[EulerPhi[9^# - 1]/# &, 25]] (* Paolo Xausa, Jun 17 2024 *)

a(n) = if(n==0, 1, eulerphi(9^n-1)/n) \\ Andrew Howroyd, Feb 01 2024

a(20) onwards from Andrew Howroyd, Feb 01 2024

A295496 a(n) = phi(6^n-1)/n, where phi is Euler's totient function (A000010).

Original entry on oeis.org

4, 12, 56, 216, 1240, 5040, 31992, 139968, 828576, 3720000, 25238048, 104509440, 803499840, 3687014016, 24373440000, 110630707200, 790546192128, 3463116249600, 25522921047520, 108957312000000, 816244048599840, 3924124012353600, 26682733370563200

Offset: 1

Seiichi Manyama, Nov 22 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Totient Function.
Wikipedia, Euler's totient function.

Column k=6 of A369291.

phi(k^n-1)/n: A011260 (k=2), A027385 (k=3), A027695 (k=4), A027741 (k=5), this sequence (k=6), A027743 (k=7), A027744 (k=8), A027745 (k=9), A295497 (k=10), A319166 (k=11).

Array[EulerPhi[6^# - 1]/# &, 25] (* Paolo Xausa, Jun 17 2024 *)

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

A295497 a(n) = phi(10^n-1)/n, where phi is Euler's totient function (A000010).

Original entry on oeis.org

6, 30, 216, 1500, 12960, 77760, 948192, 7344000, 72071856, 589032000, 6060314304, 38491200000, 496775732544, 4309959326400, 40676940288000, 345599944704000, 3921566733817776, 24555273410096640, 350877192982456140, 2915072245440000000

Offset: 1

Seiichi Manyama, Nov 22 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..50
Eric Weisstein's World of Mathematics, Totient Function.
Wikipedia, Euler's totient function.

Column k=10 of A369291.

phi(k^n-1)/n: A011260 (k=2), A027385 (k=3), A027695 (k=4), A027741 (k=5), A295496 (k=6), A027743 (k=7), A027744 (k=8), A027745 (k=9), this sequence (k=10), A319166 (k=11).

Array[EulerPhi[10^# - 1]/# &, 25] (* Paolo Xausa, Jun 17 2024 *)

PARI
```
{a(n) = eulerphi(10^n-1)/n}
```

A319166 Number of primitive polynomials of degree n over GF(11).

Original entry on oeis.org

4, 16, 144, 960, 12880, 62208, 1087632, 7027200, 85098816, 691398400, 10374307328, 49985372160, 1061265441600, 7064952935040, 90426613939200, 708867057254400, 11892871258806912, 65078340559220736, 1287559798913990448, 8819554320783360000, 111715065087913437696

Offset: 1

Seiichi Manyama, Sep 12 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..50
Eric Weisstein's World of Mathematics, Totient Function.
Wikipedia, Euler's totient function.

Column k=11 of A369291.
phi(k^n-1)/n: A011260 (k=2), A027385 (k=3), A027695 (k=4), A027741 (k=5), A295496 (k=6), A027743 (k=7), A027744 (k=8), A027745 (k=9), A295497 (k=10), this sequence (k=11).
Cf. A000010.

Array[EulerPhi[11^# - 1]/# &, 25] (* Paolo Xausa, Jun 17 2024 *)

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

a(n) = phi(11^n - 1)/n where phi is A000010.

A319213 a(n) = phi(n^3 - 1)/3 where phi is A000010.

Original entry on oeis.org

2, 4, 12, 20, 56, 36, 144, 96, 216, 144, 520, 240, 840, 480, 576, 816, 1568, 756, 2520, 1232, 1872, 1560, 4400, 1440, 4320, 3024, 4860, 3168, 7056, 2640, 9000, 5984, 7920, 6144, 10080, 4752, 17784, 7992, 13104, 9184, 22080, 7560, 23688, 12960, 14688, 15840, 33120

Offset: 2

Seiichi Manyama, Sep 13 2018

Seiichi Manyama, Table of n, a(n) for n = 2..1000
Eric Weisstein's World of Mathematics, Totient Function.
Wikipedia, Euler's totient function.

Row 3 of A369291.
Cf. A000010, A068601 (n^3-1).
phi(n^b - 1)/b: A319210 (b=2), this sequence (b=3), A319214 (b=5).

EulerPhi[Range[2, 50]^3 - 1]/3 (* Paolo Xausa, Jun 18 2024 *)

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

Sum_{k=1..n} a(k) = c * n^4 + O((n*log(n))^3), where c = (2/27) * Product_{p prime == 1 (mod 3)} (1 - 3/p^2) * Product_{p prime == 2 (mod 3)} (1 - 1/p^2) = 0.047313356295... . - Amiram Eldar, Dec 09 2024

A319210 a(n) = phi(n^2 - 1)/2 where phi is A000010.

Original entry on oeis.org

1, 2, 4, 4, 12, 8, 18, 16, 30, 16, 60, 24, 48, 48, 64, 48, 144, 48, 108, 80, 132, 80, 220, 96, 180, 144, 252, 96, 420, 128, 300, 256, 240, 192, 432, 216, 432, 288, 480, 192, 840, 240, 504, 440, 552, 352, 966, 320, 672, 480, 832, 432, 1040, 432, 720, 672, 1044, 448

Offset: 2

Seiichi Manyama, Sep 13 2018

Seiichi Manyama, Table of n, a(n) for n = 2..1000
Eric Weisstein's World of Mathematics, Totient Function.
Wikipedia, Euler's totient function.

Row 2 of A369291.
Cf. A000010, A005563 (n^2-1, shifted), A065474.
phi(n^b - 1)/b: this sequence (b=2), A319213 (b=3), A319214 (b=5).

[EulerPhi(n^2-1)/2: n in [2..70]]; // Vincenzo Librandi, Sep 15 2018

Table[(EulerPhi@(n^2 - 1) / 2), {n, 2, 70}] (* Vincenzo Librandi, Sep 15 2018 *)

PARI
```
{a(n) = eulerphi(n^2-1)/2}
```

Sum_{k=1..n} a(k) = c * n^3 / 4 + O((n*log(n))^2), where c = Product_{p prime} (1 - 2/p^2) = 0.322634... (A065474). - Amiram Eldar, Dec 09 2024

cp's OEIS Frontend

A027741 Number of primitive polynomials of degree n over GF(5).

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A027695 Number of primitive polynomials of degree n over GF(4).

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A027743 Number of primitive polynomials of degree n over GF(7).

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A027744 Number of primitive polynomials of degree n over GF(8).

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A027745 Number of primitive polynomials of degree n over GF(9).

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Maple

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A295496 a(n) = phi(6^n-1)/n, where phi is Euler's totient function (A000010).

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PARI

A295497 a(n) = phi(10^n-1)/n, where phi is Euler's totient function (A000010).

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Mathematica

PARI

A319166 Number of primitive polynomials of degree n over GF(11).

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