|
%I #41 Nov 18 2016 02:29:16
%S 2,9,25,15,21,35,33,39,65,51,45,95,69,63,161,87,93,75,217,99,185,123,
%T 117,215,141,235,329,159,105,371,177,135,305,427,201,335,213,207,245,
%U 511,189,395,165,415,581,267,261,623,1501,195,485,303,225,515,321,231
%N a(n) is the smallest number k such that cototient(k) = 2n - 1.
%C If the Goldbach conjecture holds, then for all odd numbers InvCot[2s-1] is nonempty.
%C 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 Are there infinitely many semiprimes in the sequence? - _Thomas Ordowski_, Oct 07 2016
%H Robert Israel, <a href="/A053194/b053194.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = Min{x : A051953(x)=2n-1}.
%F a(n) < (2n-1)^2 for n > 3 (if the Goldbach conjecture holds). - _Thomas Ordowski_, Oct 07 2016
%e n=18, a(18)=75, phi(75)=40, cototient(75) = 75-40 = 35 = 2*18-1.
%e 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 N:= 1000: # to get a(1) .. a(N)
%p V:= Vector(N):
%p V[1]:= 2:
%p count:= 1:
%p for k from 3 to 10^7 by 2 while count < N do
%p v:= k - numtheory:-phi(k);
%p if v::odd and v <= 2*N-1 and V[(v+1)/2] = 0 then
%p count:= count+1;
%p V[(v+1)/2]:= k;
%p fi;
%p od:
%p convert(V,list); # _Robert Israel_, Oct 10 2016
%t Table[k = 1; While[k - EulerPhi@ k != 2 n - 1, k++]; k, {n, 120}] (* _Michael De Vlieger_, Oct 10 2016 *)
%o (PARI) a(n) = k = 1; while (k - eulerphi(k) != 2*n - 1, k++); k
%Y Cf. A005278, A051953.
%K nonn
%O 1,1
%A _Labos Elemer_, Mar 02 2000
%E Name corrected by _Thomas Ordowski_, Oct 07 2016
|
