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A319099 Number of solutions to x^5 == 1 (mod n). 10
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
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).
Mobius transform gives A307380.
Sequence in context: A334561 A059592 A295554 * A098087 A354987 A361793
KEYWORD
nonn,easy,mult
AUTHOR
Jianing Song, Sep 10 2018
STATUS
approved

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Last modified March 29 03:51 EDT 2024. Contains 371264 sequences. (Running on oeis4.)