A319099 - OEIS

A319099

Number of solutions to x^5 == 1 (mod n).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 5, 1, 1, 1, 1, 1, 5, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 5, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 5, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1

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OFFSET

1,11

COMMENTS

All terms are powers of 5. Those n such that a(n) > 1 are in A066500.

LINKS

Jianing Song, Table of n, a(n) for n = 1..10000

FORMULA

Multiplicative with a(5) = 1, a(5^e) = 5 if e >= 2; for other primes p, a(p^e) = 5 if p == 1 (mod 5), a(p^e) = 1 otherwise.

If the multiplicative group of integers modulo n is isomorphic to C_{k_1} x C_{k_2} x ... x C_{k_m}, where k_i divides k_j for i < j; then a(n) = Product_{i=1..m} gcd(5, k_i).

a(n) = A000010(n)/A293482(n). - Jianing Song, Nov 10 2019

EXAMPLE

Solutions to x^5 == 1 (mod 11): x == 1, 3, 4, 5, 9 (mod 11).

Solutions to x^5 == 1 (mod 25): x == 1, 6, 11, 16, 21 (mod 25) (x == 1 (mod 5)).

Solutions to x^5 == 1 (mod 31): x == 1, 2, 4, 8, 16 (mod 31).

MATHEMATICA

f[p_, e_] := If[Mod[p, 5] == 1, 5, 1]; f[5, 1] = 1; f[5, e_] := 5; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 10 2023 *)

PROG

(PARI) a(n)=my(Z=znstar(n)[2]); prod(i=1, #Z, gcd(5, Z[i]));

CROSSREFS

Number of solutions to x^k == 1 (mod n): A060594 (k=2), A060839 (k=3), A073103 (k=4), this sequence (k=5), A319100 (k=6), A319101 (k=7), A247257 (k=8).

Cf. A030430, A066500, A293482, A000010.

Mobius transform gives A307380.

Sequence in context: A334561 A059592 A295554 * A380324 A098087 A354987

Adjacent sequences: A319096 A319097 A319098 * A319100 A319101 A319102

KEYWORD

nonn,easy,mult

AUTHOR

Jianing Song, Sep 10 2018

STATUS

approved