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

%I A319101 #25 Aug 10 2023 02:17:26
%S A319101 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 A319101 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 A319101 1,1,7,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7
%N A319101 Number of solutions to x^7 == 1 (mod n).
%C A319101 All terms are powers of 7. Those n such that a(n) > 1 are in A066502.
%H A319101 Jianing Song, <a href="/A319101/b319101.txt">Table of n, a(n) for n = 1..10000</a>
%F A319101 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 A319101 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 A319101 a(n) = A000010(n)/A293484(n). - _Jianing Song_, Nov 10 2019
%e A319101 Solutions to x^7 == 1 (mod 29): x == 1, 7, 16, 20, 23, 24, 25 (mod 29).
%e A319101 Solutions to x^7 == 1 (mod 43): x == 1, 4, 11, 16, 21, 35, 41 (mod 43).
%e A319101 Solutions to x^7 == 1 (mod 49): x == 1, 8, 15, 22, 29, 36, 43 (mod 49) (x == 1 (mod 7)).
%t A319101 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 A319101 (PARI) a(n)=my(Z=znstar(n)[2]); prod(i=1, #Z, gcd(7, Z[i]))
%Y A319101 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 A319101 Cf. A066502, A140444, A293484, A000010.
%Y A319101 Mobius transform gives A307382.
%K A319101 nonn,easy,mult
%O A319101 1,29
%A A319101 _Jianing Song_, Sep 10 2018