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A053194 a(n) is the smallest number k such that cototient(k) = 2n - 1.

%I A053194 #41 Nov 18 2016 02:29:16
%S A053194 2,9,25,15,21,35,33,39,65,51,45,95,69,63,161,87,93,75,217,99,185,123,
%T A053194 117,215,141,235,329,159,105,371,177,135,305,427,201,335,213,207,245,
%U A053194 511,189,395,165,415,581,267,261,623,1501,195,485,303,225,515,321,231
%N A053194 a(n) is the smallest number k such that cototient(k) = 2n - 1.
%C A053194 If the Goldbach conjecture holds, then for all odd numbers InvCot[2s-1] is nonempty.
%C A053194 All terms except a(1)=2 are odd numbers. All InvCototient[odd] sets seems to be nonempty, which does not hold for similar inverses of even numbers (see A005278).
%C A053194 Are there infinitely many semiprimes in the sequence? - _Thomas Ordowski_, Oct 07 2016
%H A053194 Robert Israel, <a href="/A053194/b053194.txt">Table of n, a(n) for n = 1..10000</a>
%F A053194 a(n) = Min{x : A051953(x)=2n-1}.
%F A053194 a(n) < (2n-1)^2 for n > 3 (if the Goldbach conjecture holds). - _Thomas Ordowski_, Oct 07 2016
%e A053194 n=18, a(18)=75, phi(75)=40, cototient(75) = 75-40 = 35 = 2*18-1.
%e A053194 n=12, a(12)=95 is the smallest in set {95, 119, 143, 529, ...} to the terms of which cototient(95) = cototient(119) = cototient(143) = cototient(529) = 95 - 72 = 119 - 96 = 143 - 120 = 529 - 506 = 23 = 2*12 - 1.
%p A053194 N:= 1000: # to get a(1) .. a(N)
%p A053194 V:= Vector(N):
%p A053194 V[1]:= 2:
%p A053194 count:= 1:
%p A053194 for k from 3 to 10^7 by 2 while count < N do
%p A053194   v:= k - numtheory:-phi(k);
%p A053194   if v::odd  and v <= 2*N-1 and V[(v+1)/2] = 0 then
%p A053194     count:= count+1;
%p A053194     V[(v+1)/2]:= k;
%p A053194   fi;
%p A053194 od:
%p A053194 convert(V,list); # _Robert Israel_, Oct 10 2016
%t A053194 Table[k = 1; While[k - EulerPhi@ k != 2 n - 1, k++]; k, {n, 120}] (* _Michael De Vlieger_, Oct 10 2016 *)
%o A053194 (PARI) a(n) = k = 1; while (k - eulerphi(k) != 2*n - 1, k++); k
%Y A053194 Cf. A005278, A051953.
%K A053194 nonn
%O A053194 1,1
%A A053194 _Labos Elemer_, Mar 02 2000
%E A053194 Name corrected by _Thomas Ordowski_, Oct 07 2016