A292083 Odd numbers k that can be uniquely expressed as (2*p + 1)/q for some prime p and some prime q < 2*k.

3, 7, 11, 19, 23, 37, 47, 227, 313

Offset

1,1

Comments

Conjecture: the sequence is finite and complete.

Links

Table of n, a(n) for n=1..9.

Example

313 is a member as 313 = (2*1721 + 1)/11; 11 and 1721 are prime numbers and 11 < 626.
9 is not a member since (2*13 + 1)/3 and (2*31 + 1)/7 are both valid representations where 3, 7, 13, 31 are all prime, and 3 and 7 are smaller than 18.

Mathematica

ok[n_] := Block[{p, q, s = Reduce[n == (2 p + 1)/q && q < 2 n , {p, q}, Primes]}, If[s === False, False, Length[p /. List@ ToRules@ s] == 1]]; Select[ Range[ 500] 2 + 1, ok] (* Giovanni Resta, Sep 16 2017 *)

Prog

(Magma) lst:=[]; for k in [3..313 by 2] do t:=0; for q in [3..2*k-1 by 2] do if IsPrime(q) then if IsPrime(Truncate((k*q-1)/2)) then t+:=1; if t gt 1 then break; end if; end if; if q eq PreviousPrime(2*k) and t eq 1 then Append(~lst, k); end if; end if; end for; end for; lst;
(PARI) isok(n) = {if (n % 2, nb = 0; forprime(q=2, 2*n-1, p = (n*q-1)/2; if ((denominator(p) == 1) && isprime(p), nb++); if (nb > 1, return (0)); ); nb == 1; ); }
lista(nn) = forstep(n=1, nn, 2, if (isok(n), print1(n, ", "))) \\ Michel Marcus, Sep 16 2017

Crossrefs

Cf. A233561, A292084.

Sequence in context: A118260 A018805 A191037 * A135932 A231847 A105876

Adjacent sequences: A292080 A292081 A292082 * A292084 A292085 A292086

Keyword

nonn,more

Author

Arkadiusz Wesolowski, Sep 08 2017

Status

approved