A365102 -id:A365102 - 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

A365099 Number of distinct residues of x^n (mod n^2), x=0..n^2-1.

Original entry on oeis.org

1, 2, 3, 2, 5, 4, 7, 3, 7, 6, 11, 4, 13, 8, 15, 5, 17, 8, 19, 4, 9, 12, 23, 6, 21, 14, 19, 8, 29, 12, 31, 9, 33, 18, 35, 8, 37, 20, 15, 6, 41, 8, 43, 12, 35, 24, 47, 10, 43, 22, 51, 8, 53, 20, 15, 12, 21, 30, 59, 8, 61, 32, 21, 17, 65, 24, 67, 10, 69, 24, 71, 12, 73, 38, 63

Offset: 1

Albert Mukovskiy, Aug 21 2023

nonn

Cf. A195637, A365100, A365101, A365102, A023105, A046631, A365103.

a(n) = #Set(vector(n^2, x, Mod(x-1,n^2)^n)); \\ Michel Marcus, Aug 22 2023

def A365099(n): return len({pow(x,n,n**2) for x in range(n**2)}) # Chai Wah Wu, Aug 22 2023

A365104 Number of distinct quintic residues x^5 (mod 5^n), x=0..5^n-1.

Original entry on oeis.org

1, 5, 5, 21, 101, 501, 2505, 12505, 62521, 312601, 1563001, 7815005, 39075005, 195375021, 976875101, 4884375501, 24421877505, 122109387505, 610546937521, 3052734687601, 15263673438001, 76318367190005, 381591835950005, 1907959179750021, 9539795898750101, 47698979493750501, 238494897468752505, 1192474487343762505, 5962372436718812521, 29811862183594062601

Offset: 0

Albert Mukovskiy, Aug 24 2023

It appears that for a prime p>2 the number of distinct residues x^p (mod p^n) is a(n) = (p-1)*p^(n-2) + a(n-p), with a(n<1)=1, a(1)=p.

Index entries for linear recurrences with constant coefficients, signature (5,0,0,0,1,-5).

Cf. A023105, A046631, A195637, A365099, A365100, A365101, A365102.

a[n_]:=CountDistinct[Table[PowerMod[x-1, 5, 5^(n-1)], {x, 1, 5^(n-1)}]]; Array[a, 13]

def A365104(n): return len({pow(x,5,5**n) for x in range(5**n)}) # Chai Wah Wu, Sep 17 2023

For n >= 6, a(n) = 4*5^(n-2) + a(n-5) = 5*a(n-1) + a(n-5) - 5*a(n-6). O.g.f: (-5*x^5 - 4*x^4 - 4*x^3 - 20*x^2 + 1)/(5*x^6 - x^5 - 5*x + 1). - Max Alekseyev, Feb 19 2024

Terms a(16) onward from Max Alekseyev, Feb 19 2024

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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A365099 Number of distinct residues of x^n (mod n^2), x=0..n^2-1.

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A365104 Number of distinct quintic residues x^5 (mod 5^n), x=0..5^n-1.

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