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A319099 Number of solutions to x^5 == 1 (mod n). 8
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 (list; graph; refs; listen; history; text; internal format)
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).

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 * A098087 A257099 A291578

Adjacent sequences:  A319096 A319097 A319098 * A319100 A319101 A319102

KEYWORD

nonn,easy,mult

AUTHOR

Jianing Song, Sep 10 2018

STATUS

approved

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Last modified August 4 15:53 EDT 2021. Contains 346447 sequences. (Running on oeis4.)