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

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

Author

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

Keywords

• nonn
• mult
• easy

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

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (terms to 1000 from Jordan Lenchitz)

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