A290322 - OEIS

%I #17 Jan 23 2024 08:41:54

%S 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,

%T 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,

%U 13,24,0,0,0,14,1,0,0,0,15,0,28,0,0,32,0,42,0

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

%C Conjecture: If n is divisible by 5 then a(n) > 0. - _Robert Israel_, Jan 23 2024

%H Robert Israel, <a href="/A290322/b290322.txt">Table of n, a(n) for n = 2..10000</a>

%p 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:

%t Table[Mod[Total@ Select[Range[n - 1], CoprimeQ[#, n] && CoprimeQ[Cyclotomic[5, #], n] &], n], {n, 83}] (* _Michael De Vlieger_, Jul 29 2017 *)

%o (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

%Y Cf. A058026, A289460.

%K nonn,look

%O 2,4

%A _Joshua Harrington_, _Michael Mueller_, _Jordan Lenchitz_, _Tristan Phillips_, _Madison Wellen_, _Eric Jovinelly_, Jul 27 2017