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A380604 Numbers k such that there is no number i such that A046144(i) = 2*k.

%I A380604 #15 Feb 01 2025 14:43:40
%S A380604 7,13,15,17,19,21,23,25,28,29,31,33,34,35,37,38,39,43,45,47,49,51,52,
%T A380604 53,55,57,58,59,61,62,63,67,68,69,70,71,73,74,75,76,77,79,83,85,87,91,
%U A380604 92,93,94,97,98,99,101,103,104,105,107,109,111,112,113,114,115,117,118
%N A380604 Numbers k such that there is no number i such that A046144(i) = 2*k.
%C A380604 2*a(n) are the even numbers which are not in A378508, namely numbers 2*m for which no number exists which has 2*m primitive roots. See A380594 for discussion of even numbers which are not in this sequence.
%H A380604 Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).
%H A380604 David M. Bressoud, <a href="https://www.davidbressoud.org/books-videos">A Course in Computational Number Theory (web page)</a>, <a href="https://drive.google.com/file/d/1UMesJ93mpdCabQvETMqOBH8eCR4JWA6X/view?usp=sharing">CNT.m</a>, Computational Number Theory Mathematica package.
%e A380604  There is no x such that A046144(x) = 14, so 7 is a term in this sequence (see also A380594).
%t A380604 q[n_] := Count[Join @@ PhiInverse[PhiInverse[2*n]], _?(IntegerQ[PrimitiveRoot[#]] &)] == 0; Select[Range[120], q] (* _Amiram Eldar_, Jan 28 2025, using David M. Bressoud's CNT.m *)
%o A380604 (PARI) isA033948(n) = {my(f = factor(n)); lcm(znstar(f)[2]) == eulerphi(f); }
%o A380604 isok(k) = {my(v = invphi(2*k), w, c = 0); for(i = 1, #v, c += vecsum(apply(x -> isA033948(x), invphi(v[i])))); c == 0; } \\ _Amiram Eldar_, Jan 28 2025, using _Max Alekseyev_'s invphi.gp
%Y A380604 Cf. A046144, A378508, A380594.
%K A380604 nonn
%O A380604 1,1
%A A380604 _David James Sycamore_, Jan 28 2025