A027695 -id:A027695 - cp's OEIS frontend

A011260 Number of primitive polynomials of degree n over GF(2).

1, 1, 2, 2, 6, 6, 18, 16, 48, 60, 176, 144, 630, 756, 1800, 2048, 7710, 7776, 27594, 24000, 84672, 120032, 356960, 276480, 1296000, 1719900, 4202496, 4741632, 18407808, 17820000, 69273666, 67108864, 211016256, 336849900, 929275200, 725594112, 3697909056

Offset: 1

N. J. A. Sloane

Elwyn R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84.
T. L. Booth, An analytical representation of signals in sequential networks, pp. 301-3240 of Proceedings of the Symposium on Mathematical Theory of Automata. New York, N.Y., 1962. Microwave Research Institute Symposia Series, Vol. XII; Polytechnic Press of Polytechnic Inst. of Brooklyn, Brooklyn, N.Y. 1963 xix+640 pp. See p. 303.
Pingzhi Fan and Michael Darnell, Sequence Design for Communications Applications, Wiley, NY, 1996, Table 5.1, p. 118.
W. W. Peterson and E. J. Weldon, Jr., Error-Correcting Codes. MIT Press, Cambridge, MA, 2nd edition, 1972, p. 476.
M. P. Ristenblatt, Pseudo-Random Binary Coded Waveforms, pp. 274-314 of R. S. Berkowitz, editor, Modern Radar, Wiley, NY, 1965; see p. 296.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..1206 (terms 1..400 from David W. Wilson)
Joerg Arndt, Matters Computational (The Fxtbook).
Randolph Church, Tables of irreducible polynomials for the first four prime moduli, Annals Math., 36 (1935), 198-209.
Philip Koopman, Complete lists up to N=32.
Frank Ruskey, Primitive and Irreducible Polynomials [Wayback Machine link]
Eric Weisstein's World of Mathematics, Primitive Polynomial.
Tony F. Wu, Karthik Ganesan, Yunqing Alexander Hu, H.-S. Philip Wong, Simon Wong, and Subhasish Mitra, TPAD: Hardware Trojan Prevention and Detection for Trusted Integrated Circuits, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, Vol. 35, No. 4 (2016), pp. 521-534; arXiv preprint, arXiv:1505.02211 [cs.AR], 2015.

See A058947 for initial terms.

Cf. A000010, A000020, A001037, A027695.

Maple
```
with(numtheory): phi(2^n-1)/n;
```

Table[EulerPhi[(2^n - 1)]/n, {n, 1, 50}]

a(n)=eulerphi(2^n-1)/n \\ Hauke Worpel (thebigh(AT)outgun.com), Jun 10 2008

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

A369291 Array read by antidiagonals: T(n,k) = phi(k^n-1)/n, where phi is Euler's totient function (A000010), n >= 1, k >= 2.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 4, 4, 2, 4, 4, 12, 8, 6, 2, 12, 20, 32, 22, 6, 6, 8, 56, 48, 120, 48, 18, 4, 18, 36, 216, 280, 288, 156, 16, 6, 16, 144, 160, 1240, 720, 1512, 320, 48, 4, 30, 96, 432, 1120, 5040, 5580, 4096, 1008, 60, 10, 16, 216, 640, 5400, 6048, 31992, 14976, 15552, 2640, 176

Offset: 1

Andrew Howroyd, Jan 28 2024

For k a prime power, T(n,k) is the number of primitive polynomials of degree n over GF(k). See A011260, A027385 for additional information.

			Array begins:
n\k|  2   3    4     5      6      7      8       9 ...
---+---------------------------------------------------
 1 |  1   1    2     2      4      2      6       4 ...
 2 |  1   2    4     4     12      8     18      16 ...
 3 |  2   4   12    20     56     36    144      96 ...
 4 |  2   8   32    48    216    160    432     640 ...
 5 |  6  22  120   280   1240   1120   5400    5280 ...
 6 |  6  48  288   720   5040   6048  23328   27648 ...
 7 | 18 156 1512  5580  31992  37856 254016  340704 ...
 8 | 16 320 4096 14976 139968 192000 829440 1966080 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
Eric Weisstein's World of Mathematics, Totient Function.
Wikipedia, Euler's totient function.

Rows n=1..3 and 5 are A000010(k-1), A319210, A319213, A319214.
Columns 2..11 are A011260, A027385, A027695, A027741, A295496, A027743, A027744, A027745, A295497, A319166.
Cf. A319183.

A369291[n_, k_] := EulerPhi[k^n - 1]/n;
Table[A369291[k, n-k+2], {n, 15}, {k, n}] (* Paolo Xausa, Jun 17 2024 *)

PARI
```
T(n,k) = eulerphi(k^n-1)/n
```

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.

A027742 a(n) = phi(4^n-1)/(2*n).

Original entry on oeis.org

1, 2, 6, 16, 60, 144, 756, 2048, 7776, 24000, 120032, 276480, 1719900, 4741632, 17820000, 67108864, 336849900, 725594112, 4822382628, 11842560000, 57802864896, 200778006528, 998132265920, 2283043553280, 13122000000000, 44980696051200, 178118842613760

Offset: 1

N. J. A. Sloane

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1122

Cf. A000010, A027695, A295501.

Table[EulerPhi[4^n-1]/(2n),{n,30}] (* Harvey P. Dale, May 01 2013 *)

a(n) = eulerphi(4^n-1)/(2*n) \\ Felix Fröhlich, Dec 02 2019

a(n) = A295501(n)/(2*n) = A027695(n)/2. - Amiram Eldar, Nov 30 2024

Offset corrected by Sean A. Irvine, Dec 02 2019

A319183 a(n) = phi(n^n - 1)/n where phi is A000010.

Original entry on oeis.org

1, 4, 32, 280, 5040, 37856, 829440, 15676416, 589032000, 10374307328, 388566097920, 7619466454080, 390751784579520, 11138729990400000, 575561351791902720, 24328359845627701248, 1640651748984970444800, 34709116765970413844280, 2459108342476800000000000

Offset: 2

Seiichi Manyama, Sep 12 2018

nonn

Main diagonal of the array T(n,k) = phi(n^k-1)/k for n > 1 and k > 1, which starts
    1,   2,   2,    6,     6,     18,     16, ... A011260
    2,   4,   8,   22,    48,    156,    320, ... A027385
    4,  12,  32,  120,   288,   1512,   4096, ... A027695
    4,  20,  48,  280,   720,   5580,  14976, ... A027741
   12,  56, 216, 1240,  5040,  31992, 139968, ... A295496
    8,  36, 160, 1120,  6048,  37856, 192000, ... A027743
   18, 144, 432, 5400, 23328, 254016, 829440, ... A027744

Seiichi Manyama, Table of n, a(n) for n = 2..50
Eric Weisstein's World of Mathematics, Totient Function.
Wikipedia, Euler's totient function.

A diagonal of A369291.

Cf. A000010, A006486, A027385.

Table[EulerPhi[n^n-1]/n,{n,20}] (* Harvey P. Dale, Aug 04 2020 *)

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

cp's OEIS Frontend

A011260 Number of primitive polynomials of degree n over GF(2).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A369291 Array read by antidiagonals: T(n,k) = phi(k^n-1)/n, where phi is Euler's totient function (A000010), n >= 1, k >= 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A027742 a(n) = phi(4^n-1)/(2*n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A319183 a(n) = phi(n^n - 1)/n where phi is A000010.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs