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a(n) = |{m : multiplicative order of n mod m = 9}|.
2

%I #13 Jan 26 2025 02:17:19

%S 0,2,4,26,18,12,52,4,112,12,16,12,30,12,24,488,30,24,64,4,12,78,48,28,

%T 464,12,56,62,72,12,104,56,36,52,48,112,432,28,48,52,48,24,488,56,72,

%U 288,48,24,580,6,24,116,360,44,344,16,48,104,24,8,312,44,112

%N a(n) = |{m : multiplicative order of n mod m = 9}|.

%H Alois P. Heinz, <a href="/A218258/b218258.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = tau(n^9-1)-tau(n^3-1), with tau = A000005.

%p with(numtheory):

%p a:= n-> add(mobius(9/d) *tau(n^d-1), d={3, 9}):

%p seq(a(n), n=1..80);

%t Table[DivisorSigma[0,n^9-1]-DivisorSigma[0,n^3-1],{n,70}] (* _Harvey P. Dale_, Mar 15 2018 *)

%o (PARI) a(n) = if(n == 1, 0, numdiv(n^9-1) - numdiv(n^3-1)); \\ _Amiram Eldar_, Jan 25 2025

%Y Row n=9 of A212957.

%Y Cf. A000005, A008683.

%K nonn

%O 1,2

%A _Alois P. Heinz_, Oct 24 2012