login
a(n) = |{m : multiplicative order of n mod m = 8}|.
2

%I #13 Jan 26 2025 02:19:29

%S 0,4,14,8,28,8,48,72,88,36,56,48,112,48,100,16,108,72,228,16,112,96,

%T 128,12,176,72,304,32,112,48,448,144,224,64,84,48,456,144,64,48,528,

%U 48,2064,336,152,48,800,24,300,144,228,96,608,16,704,32,256,96,688

%N a(n) = |{m : multiplicative order of n mod m = 8}|.

%H Alois P. Heinz, <a href="/A218257/b218257.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = tau(n^8-1)-tau(n^4-1), with tau = A000005.

%p with(numtheory):

%p a:= n-> add(mobius(8/d) *tau(n^d-1), d={4, 8}):

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

%t a[n_] := Subtract @@ DivisorSigma[0, {n^8-1, n^4-1}]; a[1] = 0; Array[a, 100] (* _Amiram Eldar_, Jan 25 2025 *)

%o (PARI) a(n) = if(n == 1, 0, numdiv(n^8-1) - numdiv(n^4-1)); \\ _Amiram Eldar_, Jan 25 2025

%Y Row n=8 of A212957.

%Y Cf. A000005, A008683.

%K nonn

%O 1,2

%A _Alois P. Heinz_, Oct 24 2012