A319213 -id:A319213 - 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
```

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

A319214 a(n) = phi(n^5 - 1)/5 where phi is A000010.

Original entry on oeis.org

6, 22, 120, 280, 1240, 1120, 5400, 5280, 12960, 12880, 45240, 24752, 90240, 59160, 96000, 141984, 346880, 163800, 540000, 326720, 588984, 585120, 1523280, 582400, 1728000, 1203840, 2294712, 1758096, 4692408, 1388000, 6480000, 3787200, 5416800, 4783680, 7440000

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 5 of A369291.
Cf. A000010, A258807 (n^5-1).
phi(n^b - 1)/b: A319210 (b=2), A319213 (b=3), this sequence (b=5).

Table[EulerPhi[n^5-1]/5,{n,2,40}] (* Harvey P. Dale, Feb 09 2019 *)

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

Sum_{k=1..n} a(k) = c * n^6 + O((n*log(n))^5), where c = (1/30) * Product_{p prime == 1 (mod 5)} (1 - 5/p^2) * Product_{p prime !== 1 (mod 5)} (1 - 1/p^2) = 0.019389107739... . - Amiram Eldar, Dec 09 2024

A373946 Number of primitive polynomials of third degree over GF(m) with vanishing quadratic term with m = m(n) = A000961(n), for n >= 2.

Original entry on oeis.org

1, 1, 0, 4, 3, 18, 8, 16, 18, 48, 48, 27, 80, 48, 108, 108, 72, 300, 144, 224, 180, 308, 192, 336, 560, 240, 648, 420, 576, 540, 648, 768, 1080, 1200, 912, 1360, 1008, 1352, 1188, 1584, 960, 2340, 1620, 4410, 2112, 2432, 1980, 2952, 1560, 2592, 2025, 4592, 2448, 4872, 4576

Offset: 2

Martin Becker, Jun 23 2024

nonn

Apparently, a(n) = A373514(n) * A000010( 3 * A000961(n) - 3 ) * A025474(n) / 2, for n >= 2.

			For n=5, m=5, there are 20 primitive polynomials over GF(5) of the form x^3+a*x^2+b*x+c. Among these, 4 polynomials have a=0: x^3+3*x+2, x^3+3*x+3, x^3+4*x+2, and x^3+4*x+3. Thus, a(5) = 4.

Martin Becker, Table of n, a(n) for n = 2..400

Cf. A000961, A319213, A373514.

is_max_o = (x1, x0, m, e)-> {
  for(i = 1, #e, if(x1^e[i] == x0, return(0))); x1^m == x0;
}
count_them = (q)-> {
  z = ffprimroot(ffgen(q, 'c));
  m = q^3 - 1; f = factor(m); d = #f~;
  e = vector(d, i, m/f[d + 1 - i, 1]);
  co = vector(q - 1, i, z^(i - 1));
  r = 0;
  for(a = 1, q - 1,
    for(b = 1, q - 1,
      p = co[1]*x^3 + co[a]*x + co[b];
      x1 = Mod(x, p); x0 = x1^0;
      if(is_max_o(x1, x0, m, e) && polisirreducible(p), r += 1)
    )
  );
  r;
}
print1(count_them(2));
for(q = 3, 64, if(isprimepower(q), print1(", ", count_them(q))))

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

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Magma

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A319214 a(n) = phi(n^5 - 1)/5 where phi is A000010.

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A373946 Number of primitive polynomials of third degree over GF(m) with vanishing quadratic term with m = m(n) = A000961(n), for n >= 2.

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PARI