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q[n_] := Count[(t = Table[MultiplicativeOrder[k, n], {k, Select[Range[n], CoprimeQ[n, #] &]}]), Max[t]] == EulerPhi[EulerPhi[n]]; Select[Range[200], PrimeNu[#] > 1 && q[#] &] (* Amiram Eldar, Oct 12 2021 *)

Cf. A000010, A002322, A010554, A111725.

Amiram Eldar, <a href="/A300080/b300080.txt">Table of n, a(n) for n = 1..10000</a>

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Numbers n k that are not prime powers, and with have exactly phi(phi(nk)) residues modulo n k of the maximum order.

Numbers n k with at least two distinct prime factors (A024619) such that A111725(nk) = A010554(nk).

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