A365104 -id:A365104 - cp's OEIS frontend

A365100 Number of distinct residues of x^n (mod n^3), x=0..n^3-1.

1, 3, 7, 6, 21, 8, 43, 18, 55, 22, 111, 20, 157, 44, 147, 65, 273, 56, 343, 30, 105, 112, 507, 68, 501, 158, 487, 110, 813, 88, 931, 257, 777, 274, 903, 140, 1333, 344, 371, 102, 1641, 64, 1807, 280, 1155, 508, 2163, 260, 2059, 502, 1911, 200, 2757, 488, 483, 374, 805, 814

Offset: 1

Albert Mukovskiy, Aug 21 2023

nonn

Cf. A195637, A365099, A365101, A365102, A023105, A046631, A365103, A365104.

a(n) = #Set(vector(n^3, x, Mod(x-1,n^3)^n)); \\ Michel Marcus, Aug 22 2023

def A365100(n): return len({pow(x,n,n**3) for x in range(n**3)}) # Chai Wah Wu, Aug 23 2023

A365101 Number of distinct residues of x^n (mod n^4), x=0..n^4-1.

Original entry on oeis.org

1, 4, 21, 18, 101, 30, 295, 130, 487, 153, 1211, 170, 2029, 444, 1919, 1025, 4625, 732, 6499, 442, 1881, 1818, 11639, 1290, 12501, 3045, 13123, 2516, 23549, 1530, 28831, 8193, 23009, 6939, 29795, 4148, 49285, 9750, 12863, 3354, 67241, 1500, 77659, 10302, 49187, 17460, 101615

Offset: 1

Albert Mukovskiy, Aug 21 2023

nonn

Cf. A195637, A365099, A365100, A365102, A023105, A046631, A365103, A365104.

a(n) = #Set(vector(n^4, x, Mod(x-1,n^4)^n)); \\ Michel Marcus, Aug 22 2023

def A365101(n): return len({pow(x,n,n**4) for x in range(n**4)}) # Chai Wah Wu, Aug 23 2023

A365103 Number of distinct quartic residues x^4 (mod 4^n), x=0..4^n-1.

Original entry on oeis.org

1, 2, 2, 6, 18, 70, 274, 1094, 4370, 17478, 69906, 279622, 1118482, 4473926, 17895698, 71582790, 286331154, 1145324614, 4581298450, 18325193798, 73300775186, 293203100742, 1172812402962, 4691249611846, 18764998447378

Offset: 0

Albert Mukovskiy, Aug 24 2023

nonn

a(n) = A364811(2n).
For n>=2, A319281(a(n)) == 4^n + [n mod 2 == 1].
For n>=2, a(n)=k: [ A319281(k) == 4^n + [n mod 2 == 1] ].

Index entries for linear recurrences with constant coefficients, signature (4, 1, -4).

Cf. A023105, A046631, A195637, A365104, A364811, A319281.

a[n_] = Ceiling[4^n/15] + Boole[Mod[n,2]==1]; Array[a, 24]

PARI
```
a(n) = ceil(4^n/15)+(Mod(n,2)==1);
```

def A365103(n): return len({pow(x,4,1<<(n<<1)) for x in range(1<<(n<<1))}) # Chai Wah Wu, Sep 18 2023

a(n) = ceiling(4^n/15) + (n mod 2).

A365102 Number of distinct residues of x^n (mod n^5), x=0..n^5-1.

Original entry on oeis.org

1, 7, 57, 70, 501, 140, 2059, 1029, 4377, 1255, 13311, 1820, 26365, 5150, 27555, 16386, 78609, 10940, 123463, 8190, 37785, 33280, 267675, 28700, 312501, 65915, 354295, 66950, 682893, 35140, 893731, 262145, 732105, 196525, 1031559, 142220, 1823509, 308660

Offset: 1

Albert Mukovskiy, Aug 22 2023

nonn

Cf. A195637, A365099, A365100, A365101, A365103, A365104.

a[n_]:=CountDistinct[Table[PowerMod[x-1,n,n^5],{x,0,n^5-1}]]; Array[a,38] (* Stefano Spezia, Aug 24 2023 *)

a(n) = #Set(vector(n^5, x, Mod(x-1, n^5)^n));

def A365102(n): return len({pow(x,n,n**5) for x in range(n**5)}) # Chai Wah Wu, Aug 23 2023

cp's OEIS Frontend

A365100 Number of distinct residues of x^n (mod n^3), x=0..n^3-1.

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A365101 Number of distinct residues of x^n (mod n^4), x=0..n^4-1.

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A365103 Number of distinct quartic residues x^4 (mod 4^n), x=0..4^n-1.

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A365102 Number of distinct residues of x^n (mod n^5), x=0..n^5-1.

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