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%I #35 Oct 20 2021 18:42:13
%S 1,2,3,4,5,6,7,9,10,11,13,14,15,16,17,18,19,20,22,23,25,26,27,29,30,
%T 31,32,34,35,37,38,39,40,41,43,45,46,47,48,49,50,51,52,53,54,55,58,59,
%U 60,61,62,64,67,68,70,71,73,74,75,78,79,81,82,83,85,86,87,89,90,94,95,96,97,98,100,101,102,103,104,105,106,107,109,110
%N Numbers k such that there are exactly phi(phi(k)) residues modulo k of the maximum order.
%C Numbers k such that A111725(k) = A010554(k).
%C Contains subsequences of the primes (A000040) and the prime powers (A000961) except 2^3 = 8.
%C The ratio a(n)/n tends to infinity as n grows (Müller and Schlage-Puchta, 2004).
%C Decompose (Z/kZ)* as a product of cyclic groups C_{k_1} x C_{k_2} x ... x C_{k_m}, where k_i divides k_j for i < j, then k is a term if and only if m <= 1, or v(k_{m-1},p) < v(k_m,p) holds for all primes p dividing k_m = psi(k), where v(s,p) is the p-adic valuation of s. Otherwise, there are more than phi(phi(k)) residues modulo k of the maximum order. See my Oct 12 2021 formula for A111725 for a proof. - _Jianing Song_, Oct 20 2021
%H Amiram Eldar, <a href="/A300064/b300064.txt">Table of n, a(n) for n = 1..10000</a>
%H Peter J. Cameron and D. A. Preece, <a href="https://cameroncounts.files.wordpress.com/2014/01/plr1.pdf">Primitive lambda-roots</a>, 2014.
%H T. W. Müller and J.-C. Schlage-Puchta, <a href="https://doi.org/10.4064/aa115-3-2">On the number of primitive lambda-roots</a>, Acta Arithmetica, Vol. 115 (2004), pp. 217-223.
%t q[n_] := Count[(t = Table[MultiplicativeOrder[k, n], {k, Select[Range[n], CoprimeQ[n, #] &]}]), Max[t]] == EulerPhi[EulerPhi[n]]; Select[Range[100], q] (* _Amiram Eldar_, Oct 12 2021 *)
%o (PARI) isA300064(n) = my(v=znstar(n)[2], l=#v); if(l<2, return(1), my(U=v[1], L=v[2], d=factor(U), w=omega(U)); for(i=1, w, if(valuation(L,d[i,1]) == d[i,2], return(0))); return(1)) \\ _Jianing Song_, Oct 20 2021
%Y Complement of A300065.
%Y Set union of A300079 and A000040.
%Y Set union of A300080 and A000961 \ {8}.
%Y Cf. A000010, A002322, A010554, A111725.
%K nonn
%O 1,2
%A _Max Alekseyev_, Feb 23 2018
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