A307380 - OEIS

%I #21 Feb 19 2021 03:38:05

%S 1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,

%T 0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,

%U 0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0

%N Number of quintic primitive Dirichlet characters modulo n.

%C a(n) is the number of primitive Dirichlet characters modulo n such that all entries are 0 or a fifth-power root of unity.

%C Mobius transform of A319099. Every term is 0 or a power of 4.

%H Antti Karttunen, <a href="/A307380/b307380.txt">Table of n, a(n) for n = 1..65539</a>

%F Multiplicative with a(p^e) = 4 if p^e = 25 or p == 1 (mod 5) and e = 1, otherwise 0.

%e Let w = exp(2*Pi/5). For n = 11, the 4 quintic primitive Dirichlet characters modulo n are:

%e   Chi_1 = [0, 1, w, w^3, w^2, w^4, w^4, w^2, w^3, w, 1];

%e   Chi_2 = [0, 1, w^2, w, w^4, w^3, w^3, w^4, w, w^2, 1];

%e   Chi_3 = [0, 1, w^3, w^4, w, w^2, w^2, w, w^4, w^3, 1];

%e   Chi_4 = [0, 1, w^4, w^2, w^3, w, w, w^3, w^2, w^4, 1],

%e so a(11) = 4.

%t f[5, 2] = 4; f[p_, e_] := If[Mod[p, 5] == 1 && e == 1, 4, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 16 2020 *)

%o (PARI) a(n)=sum(d=1, n, if(n%d==0, moebius(n/d)*sum(i=1, d, if((i^5-1)%d, 0, 1)), 0))

%o (PARI) A307380(n) = sumdiv(n, d, moebius(n/d)*sum(i=1, d, if((i^5-1)%d, 0, 1))); \\ (Slightly speeding the program above) - _Antti Karttunen_, Aug 22 2019

%o (PARI) A307380(n) = { my(f=factor(n)); prod(i=1, #f~, if(((5==f[i,1])&&(2==f[i,2]))||((1==(f[i,1]%5))&&(1==f[i,2])),4,0)); }; \\ (After the multiplicative formula, much faster) - _Antti Karttunen_, Aug 22 2019

%Y Number of k-th power primitive Dirichlet characters modulo n: A114643 (k=2), A160498 (k=3), A160499 (k=4), this sequence (k=5), A307381 (k=6), A307382 (k=7), A329272 (k=8).

%Y Cf. A319099 (number of solutions to x^5 == 1 (mod n)).

%K nonn,mult

%O 1,11

%A _Jianing Song_, Apr 06 2019

%E More terms from _Antti Karttunen_, Aug 22 2019