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A218259 a(n) = |{m : multiplicative order of n mod m = 10}|.

%I A218259 #10 Jan 26 2025 02:16:09
%S A218259 0,5,16,38,47,40,66,84,150,33,72,10,106,178,168,140,265,54,534,106,68,
%T A218259 20,296,56,564,231,732,52,730,26,604,114,80,164,348,332,297,50,912,84,
%U A218259 904,110,88,130,222,552,332,76,1430,81,702,236,950,178,256,168,564
%N A218259 a(n) = |{m : multiplicative order of n mod m = 10}|.
%H A218259 Alois P. Heinz, <a href="/A218259/b218259.txt">Table of n, a(n) for n = 1..10000</a>
%F A218259 a(n) = tau(n^10-1)-tau(n^5-1)-tau(n^2-1)+tau(n-1), with tau = A000005.
%p A218259 with(numtheory):
%p A218259 a:= n-> add(mobius(10/d) *tau(n^d-1), d={1, 2, 5, 10}):
%p A218259 seq(a(n), n=1..80);
%t A218259 a[n_] := Total[{1, -1, -1, 1} * DivisorSigma[0, n^{10, 5, 2, 1} - 1]]; a[1] = 0; Array[a, 100] (* _Amiram Eldar_, Jan 25 2025*)
%o A218259 (PARI) a(n) = if(n == 1, 0, numdiv(n^10-1) - numdiv(n^5-1) - numdiv(n^2-1) + numdiv(n-1)); \\ _Amiram Eldar_, Jan 25 2025
%Y A218259 Row n=10 of A212957.
%Y A218259 Cf. A000005, A008683.
%K A218259 nonn
%O A218259 1,2
%A A218259 _Alois P. Heinz_, Oct 24 2012