A319101 - OEIS

%I #25 Aug 10 2023 02:17:26

%S 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,

%T 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,

%U 1,1,7,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7

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

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

%H Jianing Song, <a href="/A319101/b319101.txt">Table of n, a(n) for n = 1..10000</a>

%F 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.

%F 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).

%F a(n) = A000010(n)/A293484(n). - _Jianing Song_, Nov 10 2019

%e Solutions to x^7 == 1 (mod 29): x == 1, 7, 16, 20, 23, 24, 25 (mod 29).

%e Solutions to x^7 == 1 (mod 43): x == 1, 4, 11, 16, 21, 35, 41 (mod 43).

%e Solutions to x^7 == 1 (mod 49): x == 1, 8, 15, 22, 29, 36, 43 (mod 49) (x == 1 (mod 7)).

%t 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 *)

%o (PARI) a(n)=my(Z=znstar(n)[2]); prod(i=1, #Z, gcd(7, Z[i]))

%Y 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).

%Y Cf. A066502, A140444, A293484, A000010.

%Y Mobius transform gives A307382.

%K nonn,easy,mult

%O 1,29

%A _Jianing Song_, Sep 10 2018