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A053194 a(n) is the smallest number k such that cototient(k) = 2n - 1. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

LINKS

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

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

EXAMPLE

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.

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 *)

PROG

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

CROSSREFS

Cf. A005278, A051953.

Sequence in context: A097346 A261431 A226388 * A005582 A173965 A116454

Adjacent sequences:  A053191 A053192 A053193 * A053195 A053196 A053197

KEYWORD

nonn

AUTHOR

Labos Elemer, Mar 02 2000

EXTENSIONS

Name corrected by Thomas Ordowski, Oct 07 2016

STATUS

approved

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Last modified August 12 17:17 EDT 2020. Contains 336439 sequences. (Running on oeis4.)