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%I #27 Aug 10 2023 02:17:48
%S 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,
%T 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,
%U 1,1,5,1,1,1,5,1,5,1,1,1,1,5,1,1,1,1,1
%N Number of solutions to x^5 == 1 (mod n).
%C All terms are powers of 5. Those n such that a(n) > 1 are in A066500.
%H Jianing Song, <a href="/A319099/b319099.txt">Table of n, a(n) for n = 1..10000</a>
%F 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.
%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(5, k_i).
%F a(n) = A000010(n)/A293482(n). - _Jianing Song_, Nov 10 2019
%e Solutions to x^5 == 1 (mod 11): x == 1, 3, 4, 5, 9 (mod 11).
%e Solutions to x^5 == 1 (mod 25): x == 1, 6, 11, 16, 21 (mod 25) (x == 1 (mod 5)).
%e Solutions to x^5 == 1 (mod 31): x == 1, 2, 4, 8, 16 (mod 31).
%t 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 *)
%o (PARI) a(n)=my(Z=znstar(n)[2]); prod(i=1,#Z,gcd(5,Z[i]));
%Y 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).
%Y Cf. A030430, A066500, A293482, A000010.
%Y Mobius transform gives A307380.
%K nonn,easy,mult
%O 1,11
%A _Jianing Song_, Sep 10 2018
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