A292083 - OEIS

%I #35 Sep 08 2022 08:46:19

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

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

%C Conjecture: the sequence is finite and complete.

%e 313 is a member as 313 = (2*1721 + 1)/11; 11 and 1721 are prime numbers and 11 < 626.

%e 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.

%t 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 *)

%o (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;

%o (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;);}

%o lista(nn) = forstep(n=1, nn, 2, if (isok(n), print1(n, ", "))) \\ _Michel Marcus_, Sep 16 2017

%Y Cf. A233561, A292084.

%K nonn,more

%O 1,1

%A _Arkadiusz Wesolowski_, Sep 08 2017