A295496 -id:A295496 - cp's OEIS frontend

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.

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
```

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.

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

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.

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

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

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

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