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A307380 Number of quintic primitive Dirichlet characters modulo n. 5
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,11

COMMENTS

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

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

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65539

FORMULA

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

EXAMPLE

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

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

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

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

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

so a(11) = 4.

PROG

(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))

(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

(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

CROSSREFS

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).

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

Sequence in context: A061858 A005873 A217511 * A178990 A276570 A236378

Adjacent sequences:  A307377 A307378 A307379 * A307381 A307382 A307383

KEYWORD

nonn,mult

AUTHOR

Jianing Song, Apr 06 2019

EXTENSIONS

More terms from Antti Karttunen, Aug 22 2019

STATUS

approved

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Last modified January 26 10:40 EST 2020. Contains 331279 sequences. (Running on oeis4.)