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

2, 9, 25, 15, 21, 35, 33, 39, 65, 51, 45, 95, 69, 63, 161, 87, 93, 75, 217, 99, 185, 123, 117, 215, 141, 235, 329, 159, 105, 371, 177, 135, 305, 427, 201, 335, 213, 207, 245, 511, 189, 395, 165, 415, 581, 267, 261, 623, 1501, 195, 485, 303, 225, 515, 321, 231

Offset: 1

Author

Labos Elemer, Mar 02 2000

Keywords

• nonn

Comments

If the Goldbach conjecture holds, then for all odd numbers InvCot[2s-1] is nonempty.

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).

Are there infinitely many semiprimes in the sequence? - Thomas Ordowski, Oct 07 2016

Examples

			n=18, a(18)=75, phi(75)=40, cototient(75) = 75-40 = 35 = 2*18-1.
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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005278, A051953.

Programs

Maple

N:= 1000: # to get a(1) .. a(N)
V:= Vector(N):
V[1]:= 2:
count:= 1:
for k from 3 to 10^7 by 2 while count < N do
  v:= k - numtheory:-phi(k);
  if v::odd  and v <= 2*N-1 and V[(v+1)/2] = 0 then
    count:= count+1;
    V[(v+1)/2]:= k;
  fi;
od:
convert(V,list); # Robert Israel, Oct 10 2016

Mathematica

Table[k = 1; While[k - EulerPhi@ k != 2 n - 1, k++]; k, {n, 120}] (* Michael De Vlieger, Oct 10 2016 *)

PARI

a(n) = k = 1; while (k - eulerphi(k) != 2*n - 1, k++); k

Formula

a(n) = Min{x : A051953(x)=2n-1}.

a(n) < (2n-1)^2 for n > 3 (if the Goldbach conjecture holds). - Thomas Ordowski, Oct 07 2016

Extensions

Name corrected by Thomas Ordowski, Oct 07 2016