This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
lst={};Do[p=Prime[n];If[ !PrimeQ[Floor[p/2]+p]&&!PrimeQ[Ceiling[p/2]+p],AppendTo[lst,p]],{n,6!}];lst
The POBAs for 1 start with 3/2, 2/3, 5/7, 13/11, 11/13, 19/17, 17/19, 31/29, 29/31, 43/41, 41/43, 61/59, 59/61. For example, if p and q are primes and q > 13, then 11/13 is closer to 1 than p/q is.
x = 1; z = 200; p[k_] := p[k] = Prime[k];
t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265759/A265760 *)
Numerator[tL] (* A001359 *)
Denominator[tL] (* A006512 *)
Numerator[tU] (* A006512 *)
Denominator[tU] (* A001359 *)
Numerator[y] (* A265759 *)
Denominator[y] (* A265760 *)
lst={};Do[p=Prime[n];If[PrimeQ[p=Floor[p/2]+p],If[PrimeQ[p=Floor[p/2]+p],AppendTo[lst,Prime[n]]]],{n,7!}];lst
prQ[n_]:=Module[{p1=Floor[n/2]+n},AllTrue[{p1,Floor[p1/2]+p1},PrimeQ]]; Select[Prime[Range[1600]],prQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 21 2016 *)
lst={};Do[p=Prime[n];If[PrimeQ[p=Floor[p/2]+p],If[PrimeQ[p=Ceiling[p/2]+p],AppendTo[lst,Prime[n]]]],{n,7!}];lst
pQ[n_]:=Module[{p1=Floor[n/2]+n},AllTrue[{p1,Ceiling[p1/2]+p1},PrimeQ]]; Select[Prime[Range[2500]],pQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 07 2019 *)
67 is in the sequence because 67, ceiling(67/2) + 67 = 101 and floor(101/2) + 101 = 151 are all primes.
N:= 10^5; # to get all entries <= N
filter:= proc(p)
local p1,p2;
if not isprime(p) then return false fi;
p1:= ceil(p/2)+p;
if not isprime(p1) then return false fi;
p2:= floor(p1/2)+p1;
isprime(p2);
end proc;
select(filter,[seq(2*i+1,i=1..floor((N-1)/2)]; # Robert Israel, May 09 2014
lst={};Do[p=Prime[n];If[PrimeQ[p=Ceiling[p/2]+p],If[PrimeQ[p=Floor[p/2]+p],AppendTo[lst,Prime[n]]]],{n,7!}];lst
cpQ[n_]:=Module[{p1=Ceiling[n/2]+n},AllTrue[{p1,Ceiling[p1/2]+n}, PrimeQ]]; Select[Prime[Range[2000]],cpQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 29 2017 *)
3+2=5, 7+4=11, 11+6=17, 19+10=29, 31+16=47, 47+24=71,.. r:3,7,11,19,31,47,59,67,71,127,131,151,167,179,211,239,307,311, ..A158709.
filter:= p -> isprime(p) and isprime((2*p-1)/3): select(filter, [seq(i,i=5..10000,6)]); # Robert Israel, Aug 19 2020
lst={};Do[r=Prime[n];p=r+(r+1)/2;If[PrimeQ[p],AppendTo[lst,p]],{n,6!}];lst
Select[#+(#+1)/2&/@Prime[Range[300]],PrimeQ] (* Harvey P. Dale, Apr 30 2015 *)
lst={};Do[p=Prime[n];If[ !PrimeQ[p1=Floor[p/2]+p]&&!PrimeQ[p2=Ceiling[p/2]+p],If[ !PrimeQ[p3=Floor[p1/2]+p1]&&!PrimeQ[p5=Ceiling[p1/2]+p1],If[ !PrimeQ[p4=Floor[p2/2]+p2]&&!PrimeQ[p6=Ceiling[p2/2]+p2],AppendTo[lst,Prime[n]]]]],{n,6!}];lst
nonpQ[p_]:=Module[{p1=Floor[p/2]+p,p2=Ceiling[p/2]+p},NoneTrue[ {p1,p2,Floor[ p1/2]+p1,Ceiling[p1/2]+p1,Floor[p2/2]+p2,Ceiling[p2/2]+ p2},PrimeQ]]; Select[Prime[Range[200]],nonpQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 21 2019 *)
lst={};Do[p=Prime[n];If[PrimeQ[Floor[p/3]+p],AppendTo[lst,p]],{n,6!}];lst
2 is in the sequence because (2 + 1)/3 + 2 = 1 + 2 = 3, which is prime. 5 is in the sequence because (5 + 1)/3 + 5 = 2 + 5 = 7, which is prime. 11 is not in the sequence because (11 + 1)/3 + 11 = 15 = 3 * 5.
Select[Prime[Range[350]], PrimeQ[(# + 1)/3 + #] &] (* Harvey P. Dale, Feb 24 2013, simplified by Alonso del Arte, Mar 12 2016 *)
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