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[Ceiling[p/2]+p],AppendTo[lst,p]],{n,6!}];lst
Select[Prime@ Range@ 250, PrimeQ@ Ceiling[3#/2] &] (* Vincenzo Librandi, Apr 15 2013 and slightly modified by Robert G. Wilson v, Feb 26 2018 *)
forprime(p=2,1e4,if(isprime(p+ceil(p/2)),print1(p", "))) \\ Charles R Greathouse IV, Nov 09 2011
print1(2);forprime(p=3,1e4,if(p%4==3&&isprime(p\4*6+5),print1(", "p))) \\ Charles R Greathouse IV, Nov 09 2011
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 *)
7 is in the sequence because (2 * 7 + 1)/3 = 5, which is also prime. 19 is in the sequence because (2 * 19 + 1)/3 = 13, which is also prime.
Select[Range[7, 2539, 2], PrimeQ[#] && PrimeQ[(2# + 1)/3]&] (* Zak Seidov, Jul 31 2012 *) Select[Prime[Range[400]], PrimeQ[(2 # + 1) / 3]&] (* Vincenzo Librandi, Apr 14 2013 *)
is(n)=n%3==1 && isprime((2*n+1)/3) && isprime(n) \\ Charles R Greathouse IV, Jul 31 2012
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 *)
p=2 qualifies since 2*1+1=3 is prime. p=5 qualifies since 5*2+1=11 is prime.
lst={};Do[p=Prime[n];If[PrimeQ[p*Floor[p/2]+1],AppendTo[lst,p]],{n,3*6!}]; lst
Select[Prime[Range[350]],PrimeQ[1+#*Floor[#/2]]&] (* Harvey P. Dale, Apr 07 2015 *)
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