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.

User: Michael Mueller

Michael Mueller's wiki page.

Michael Mueller has authored 5 sequences.

A290322 Sum modulo n of all units u in Z/nZ such that Phi(5,u) is a unit, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

1, 0, 0, 4, 0, 0, 0, 0, 9, 1, 0, 0, 0, 3, 0, 0, 0, 0, 8, 0, 12, 0, 0, 20, 0, 0, 0, 0, 18, 1, 0, 24, 0, 14, 0, 0, 0, 0, 16, 1, 0, 0, 24, 9, 0, 0, 0, 0, 45, 0, 0, 0, 0, 14, 0, 0, 0, 0, 36, 1, 32, 0, 0, 13, 24, 0, 0, 0, 14, 1, 0, 0, 0, 15, 0, 28, 0, 0, 32, 0, 42, 0
Offset: 2

Views

Author

Joshua Harrington, Michael Mueller, Jordan Lenchitz, Tristan Phillips, Madison Wellen, Eric Jovinelly, Jul 27 2017

Keywords

Comments

Conjecture: If n is divisible by 5 then a(n) > 0. - Robert Israel, Jan 23 2024

Links

Crossrefs

Cf. A058026, A289460.

Programs

  • Maple
    with(numtheory): m:=5: for n from 2 to 100 do S:={}: for a from 1 to n-1 do if gcd(a,n)=1 and gcd(cyclotomic(m,a),n)=1 then S:={op(S),a}: fi: od: print(sum(op(i,S),i=1..nops(S)) mod n): od:
  • Mathematica
    Table[Mod[Total@ Select[Range[n - 1], CoprimeQ[#, n] && CoprimeQ[Cyclotomic[5, #], n] &], n], {n, 83}] (* Michael De Vlieger, Jul 29 2017 *)
  • PARI
    a(n) = sum(k=0, n-1, k*((gcd(n, k)==1) && (gcd(n, polcyclo(5, k))==1))) % n; \\ Michel Marcus, Jul 29 2017

A290321 Sum modulo n of all units u in Z/nZ such that Phi(3,u) is a unit, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

1, 2, 0, 0, 5, 1, 0, 6, 0, 0, 4, 1, 8, 5, 0, 0, 15, 1, 0, 8, 0, 0, 8, 0, 14, 18, 16, 0, 20, 1, 0, 11, 0, 25, 12, 1, 20, 14, 0, 0, 8, 1, 0, 15, 0, 0, 16, 7, 0, 17, 28, 0, 45, 0, 32, 20, 0, 0, 40, 1, 32, 24, 0, 30, 44, 1, 0, 23, 60, 0, 24, 1, 38, 25, 40, 66, 14, 1
Offset: 2

Views

Author

Michael Mueller, Jordan Lenchitz, Tristan Phillips, Madison Wellen, Eric Jovinelly, Joshua Harrington, Jul 27 2017

Keywords

Crossrefs

Cf. A058026, A289460.

Programs

  • Maple
    with(numtheory): m:=3: for n from 2 to 100 do S:={}: for a from 1 to n-1 do if gcd(a,n)=1 and gcd(cyclotomic(m,a),n)=1 then S:={op(S),a}: fi: od: print(sum(op(i,S),i=1..nops(S)) mod n): od:
  • Mathematica
    Table[Mod[Total@ Select[Range[n - 1], CoprimeQ[#, n] && CoprimeQ[Cyclotomic[3, #], n] &], n], {n, 79}] (* Michael De Vlieger, Jul 29 2017 *)
  • PARI
    a(n) = sum(k=0, n-1, k*((gcd(n, k)==1) && (gcd(n, polcyclo(3, k))==1))) % n; \\ Michel Marcus, Jul 29 2017

A289835 Number of units u in Z/(2n-1)Z such that Phi(4,u) is a unit, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

1, 2, 2, 6, 6, 10, 10, 4, 14, 18, 12, 22, 10, 18, 26, 30, 20, 12, 34, 20, 38, 42, 12, 46, 42, 28, 50, 20, 36, 58, 58, 36, 20, 66, 44, 70, 70, 20, 60, 78, 54, 82, 28, 52, 86, 60, 60, 36, 94, 60, 98, 102, 24, 106, 106, 68, 110, 44, 60, 84, 110, 76, 50, 126, 84
Offset: 1

Views

Author

Jordan Lenchitz, Michael Mueller, Tristan Phillips, Madison Wellen, Eric Jovinelly, Joshua Harrington, Jul 25 2017

Keywords

Comments

If k is even, the number of units u in Z/kZ such that Phi(4,u) is a unit is zero.

Links

Crossrefs

Cf. A058026, A289460.

Programs

  • Maple
    m:=4 do for t from 1 to 1000 do n:=2*t-1: S:={}: for a from 0 to n-1 do if gcd(a,n)=1 and gcd(cyclotomic(m,a),n)=1 then S:={op(S),a}: fi: od: print(t,nops(S)): od: od:
  • PARI
    a(n) = sum(k=0, 2*n-2, (gcd(2*n-1, k)==1) && (gcd(2*n-1, polcyclo(4, k))==1)); \\ Michel Marcus, Jul 29 2017

A290309 Number of units u in Z/nZ such that Phi(5,u) is a unit, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 6, 4, 6, 3, 6, 4, 12, 6, 6, 8, 16, 6, 18, 6, 12, 6, 22, 8, 15, 12, 18, 12, 28, 6, 26, 16, 12, 16, 18, 12, 36, 18, 24, 12, 36, 12, 42, 12, 18, 22, 46, 16, 42, 15, 32, 24, 52, 18, 18, 24, 36, 28, 58, 12, 56, 26, 36, 32, 36, 12, 66, 32, 44, 18
Offset: 1

Views

Author

Tristan Phillips, Jordan Lenchitz, Michael Mueller, Madison Wellen, Eric Jovinelly, Joshua Harrington, Jul 27 2017

Keywords

Comments

If n is a prime other than 5, then a(n) = n - 5 if n == 1 (mod 10), otherwise a(n) = n - 1. - Robert Israel, Jul 31 2017

Links

Crossrefs

Cf. A058026 (with Phi(1,u) or Phi(2,u)), A289460 (with Phi(3,u)).

Programs

  • Maple
    m:=5; T:=[]: for n from 1 to 100 do S:={}: for a from 0 to n-1 do if gcd(a,n)=1 and gcd(cyclotomic(m,a),n)=1 then S:={op(S),a}: fi: od: T:=[op(T),nops(S)]: od: print(T):
  • Mathematica
    Table[Count[Range[n - 1], k_ /; And[CoprimeQ[k, n], CoprimeQ[Cyclotomic[5, k], n]]], {n, 70}] (* Michael De Vlieger, Jul 30 2017 *)
  • PARI
    a(n) = sum(k=0, n-1, (gcd(n, k)==1) && (gcd(n, polcyclo(5, k))==1)); \\ Michel Marcus, Jul 29 2017

A289460 Number of units u in Z/nZ such that Phi(3,u) is a unit, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

1, 1, 1, 2, 4, 1, 4, 4, 3, 4, 10, 2, 10, 4, 4, 8, 16, 3, 16, 8, 4, 10, 22, 4, 20, 10, 9, 8, 28, 4, 28, 16, 10, 16, 16, 6, 34, 16, 10, 16, 40, 4, 40, 20, 12, 22, 46, 8, 28, 20, 16, 20, 52, 9, 40, 16, 16, 28, 58, 8, 58, 28, 12, 32, 40, 10, 64, 32, 22, 16, 70, 12
Offset: 1

Views

Author

Jordan Lenchitz, Michael Mueller, Tristan Phillips, Madison Wellen, Eric Jovinelly, Joshua Harrington, Jul 06 2017

Keywords

Comments

The number of units u in Z/nZ such that Phi(1,u) or Phi(2,u) is a unit is given by A058026.

Links

Crossrefs

Cf. A058026.

Programs

  • Maple
    m:=3; T:=[]: for n from 2 to 1000 do S:={}: for a from 0 to n-1 do if gcd(a,n)=1 and gcd(cyclotomic(m,a),n)=1 then S:={op(S),a}: fi: od: T:=[op(T),nops(S)]: od: print(m,T):
  • Mathematica
    Table[Count[Map[Cyclotomic[3, #] &, Select[Range@ n, CoprimeQ[#, n] &]], u_ /; CoprimeQ[u, n]], {n, 72}] (* Michael De Vlieger, Jul 11 2017 *)
  • PARI
    g(n)=sum(k=0,n-1, gcd(k,n)==1 && gcd(polcyclo(3,k),n)==1)
a(n)=my(f=factor(n)); prod(i=1,#f~, g(f[i,1]^f[i,2])) \\ Charles R Greathouse IV, Jul 06 2017

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