cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Original entry on oeis.org

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

Views

Author

Albert Mukovskiy, Aug 21 2023

Keywords

Crossrefs

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

Programs

  • PARI
    a(n) = #Set(vector(n^3, x, Mod(x-1,n^3)^n)); \\ Michel Marcus, Aug 22 2023
  • Python
    def A365100(n): return len({pow(x,n,n**3) for x in range(n**3)}) # 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

Views

Author

Albert Mukovskiy, Aug 21 2023

Keywords

Crossrefs

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

Programs

  • PARI
    a(n) = #Set(vector(n^2, x, Mod(x-1,n^2)^n)); \\ Michel Marcus, Aug 22 2023
  • Python
    def A365099(n): return len({pow(x,n,n**2) for x in range(n**2)}) # Chai Wah Wu, Aug 22 2023

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

Views

Author

Albert Mukovskiy, Aug 22 2023

Keywords

Crossrefs

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

Programs

  • Mathematica
    a[n_]:=CountDistinct[Table[PowerMod[x-1,n,n^5],{x,0,n^5-1}]]; Array[a,38] (* Stefano Spezia, Aug 24 2023 *)
  • PARI
    a(n) = #Set(vector(n^5, x, Mod(x-1, n^5)^n));
  • Python
    def A365102(n): return len({pow(x,n,n**5) for x in range(n**5)}) # Chai Wah Wu, Aug 23 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

Views

Author

Albert Mukovskiy, Aug 24 2023

Keywords

Comments

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.

Links

Crossrefs

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

Programs

  • Mathematica
    a[n_]:=CountDistinct[Table[PowerMod[x-1, 5, 5^(n-1)], {x, 1, 5^(n-1)}]]; Array[a, 13]
  • Python
    def A365104(n): return len({pow(x,5,5**n) for x in range(5**n)}) # Chai Wah Wu, Sep 17 2023

Formula

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

Extensions

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

A366420 Number of distinct integers of the form (x^n + y^n) mod n^4.

Original entry on oeis.org

1, 9, 45, 35, 325, 95, 931, 259, 1215, 625, 6171, 627, 12337, 2210, 14625, 2049, 32657, 2435, 58843, 1683, 11025, 12105, 140185, 4883, 40625, 16055, 32805, 14586, 236321, 11875, 375751, 16385, 277695, 59245, 302575, 16071, 789913, 97475, 98865, 13107, 1413721, 9405, 1399693
Offset: 1

Views

Author

Albert Mukovskiy, Oct 11 2023

Keywords

Comments

It is enough to take x,y from {0,1,...,n^3-1}.
It appears that the number of distinct integers of the form x^(p^k) + y^(p^k) (mod (p^k)^m) for a prime p>2 and natural k is A121278(p)*p^(k-1)*p^(k*(m-2)) for m>1. For m=1 see A366418.
It appears that the number of distinct integers of the form x^n + y^n (mod n^m) for an odd n, m>1 is A121278(n)*n^(m-2).

Crossrefs

Cf. A366418, A121278, A366419, A365101.

Programs

  • PARI
    a(n) = #setbinop((x, y)->Mod(x, n^4)^n+Mod(y, n^4)^n, [0..n^3-1]); \\ Michel Marcus, Oct 14 2023
  • Python
    def A366420(n):
    m = n**4
    return len({(pow(x,n,m)+pow(y,n,m))%m for x in range(n**3) for y in range(x+1)}) # Chai Wah Wu, Nov 12 2023
Showing 1-5 of 5 results.

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