A319101 Number of solutions to x^7 == 1 (mod n).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 7

Offset

1,29

Comments

All terms are powers of 7. Those n such that a(n) > 1 are in A066502.

Links

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

Formula

Multiplicative with a(7) = 1, a(7^e) = 7 if e >= 2; for other primes p, a(p^e) = 7 if p == 1 (mod 7), 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(7, k_i).

a(n) = A000010(n)/A293484(n). - Jianing Song, Nov 10 2019

Example

Solutions to x^7 == 1 (mod 29): x == 1, 7, 16, 20, 23, 24, 25 (mod 29).
Solutions to x^7 == 1 (mod 43): x == 1, 4, 11, 16, 21, 35, 41 (mod 43).
Solutions to x^7 == 1 (mod 49): x == 1, 8, 15, 22, 29, 36, 43 (mod 49) (x == 1 (mod 7)).

Mathematica

f[p_, e_] := If[Mod[p, 7] == 1, 7, 1]; f[7, 1] = 1; f[7, e_] := 7; 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(7, Z[i]))

Crossrefs

Number of solutions to x^k == 1 (mod n): A060594 (k=2), A060839 (k=3), A073103 (k=4), A319099 (k=5), A319100 (k=6), this sequence (k=7), A247257 (k=8).

Cf. A066502, A140444, A293484, A000010.

Mobius transform gives A307382.

Sequence in context: A348970 A053878 A070672 * A354487 A336699 A239700

Adjacent sequences: A319098 A319099 A319100 * A319102 A319103 A319104

Keyword

nonn,easy,mult

Author

Jianing Song, Sep 10 2018

Status

approved