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.
a002822 n = a002822_list !! (n-1)
a002822_list = f a000040_list where
f (q:ps'@(p:ps)) | p > q + 2 || r > 0 = f ps'
| otherwise = y : f ps where (y,r) = divMod (q + 1) 6
-- Reinhard Zumkeller, Jul 13 2014
[n: n in [1..200] | IsPrime(6*n+1) and IsPrime(6*n-1)] // Vincenzo Librandi, Nov 21 2010
select(n -> isprime(6*n-1) and isprime(6*n+1), [$1..1000]); # Robert Israel, Jan 11 2015
Select[ Range[350], PrimeQ[6# - 1] && PrimeQ[6# + 1] & ]
Select[Range[400],AllTrue[6#+{1,-1},PrimeQ]&] (* Harvey P. Dale, Jul 27 2022 *)
#/6&/@Select[Range[6,2500,6],AllTrue[#+{1,-1},PrimeQ]&] (* Harvey P. Dale, Mar 31 2023 *)
select(primes(100),n->isprime(n-2)&&n>5)\6 \\ Charles R Greathouse IV, Jul 05 2011
p=5; forprime(q=5, 1e4, if(q-p==2, print1((p+1)/6", ")); p=q); \\ Altug Alkan, Oct 13 2015
list(lim)=my(v=List(),p=5); forprime(q=7,6*lim+1, if(q-p==2, listput(v,q\6)); p=q); Vec(v) \\ Charles R Greathouse IV, Dec 03 2016
a(2)=8 because 6*8 + 1 = 49, which is composite.
Filtered([0..250], k-> not IsPrime(6*k+1)) # G. C. Greubel, Feb 21 2019
a046954 n = a046954_list !! (n-1) a046954_list = map (`div` 6) $ filter ((== 0) . a010051' . (+ 1)) [0,6..] -- Reinhard Zumkeller, Jul 13 2014
[n: n in [0..250] | not IsPrime(6*n+1)]; // G. C. Greubel, Feb 21 2019
remove(k-> isprime(6*k+1), [$0..140])[]; # Muniru A Asiru, Feb 22 2019
a = Flatten[Table[If[PrimeQ[6*n + 1] == False, n, {}], {n, 0, 50}]] (* Roger L. Bagula, May 17 2007 *)
Select[Range[0, 200], !PrimeQ[6 # + 1] &] (* Vincenzo Librandi, Sep 27 2013 *)
is(n)=!isprime(6*n+1) \\ Charles R Greathouse IV, Aug 01 2016
[n for n in (0..250) if not is_prime(6*n+1)] # G. C. Greubel, Feb 21 2019
For n = 1, 2 and 3, there are no nonzero x,y such that n = |6xy + x + y|, and (6n-1, 6n+1) = (5, 7), (11, 13) and (17, 19) are indeed twin primes. For n = 4 we have x = y = -1 such that |6xy + x + y| = |6 - 1 - 1| = 4 and (23, 25) is indeed not a twin prime pair.
f:= proc(n) local V, C, t, m,v, r;
V:= numtheory:-divisors(6*n+1) minus {1,6*n+1};
C:= map(u -> `if`(u mod 6 = 1, [(u-1)/6, ((6*n+1)/u - 1)/6], [(-u-1)/6, (-(6*n+1)/u - 1)/6]), V);
V:= numtheory:-divisors(6*n-1) minus {1,6*n-1};
C:= C union map(u -> `if`(u mod 6 = 1, [(u-1)/6, ((-6*n+1)/u - 1)/6], [(-u-1)/6, ((-6*n+1)/u - 1)/6]), V);
C:= select(t -> abs(t[1]) >= abs(t[2]), C)[..,1];
if C = {} then return 0 fi;
m:= infinity;
for t in C do
if abs(t) < m then m:= abs(t); r:= t;
elif abs(t) = m and t > 0 then r:= t
fi
od;
r
end proc:
map(f, [$1..100]); # Robert Israel, Jul 21 2025
{A384102(n)=for(x=1,n\/5, my(p=6*x+1, q=6*x-1, r=if((n-x)%p==0, (n-x)\p, (n+x)%p==0, (n+x)\p, (n-x)%q==0, (x-n)\q, (n+x)%q==0,-(n+x)\q)); r && abs(r) <= x && return(sign(r)*x))}
For n = 1, 2 and 3, there are no nonzero x,y such that n = |6xy + x + y|, and (6n-1, 6n+1) = (5, 7), (11, 13) and (17, 19) are indeed twin primes. For n = 4 we have x = y = -1 such that |6xy + x + y| = |6 - 1 - 1| = 4 and (23, 25) is indeed not a twin prime pair.
f:= proc(n) local V, C, t, m, v, r;
V:= numtheory:-divisors(6*n+1) minus {1, 6*n+1};
C:= map(u -> `if`(u mod 6 = 1, [(u-1)/6, ((6*n+1)/u - 1)/6], [(-u-1)/6, (-(6*n+1)/u - 1)/6]), V);
V:= numtheory:-divisors(6*n-1) minus {1, 6*n-1};
C:= C union map(u -> `if`(u mod 6 = 1, [(u-1)/6, ((-6*n+1)/u - 1)/6], [(-u-1)/6, ((6*n-1)/u - 1)/6]), V);
C:= select(t -> abs(t[1]) >= abs(t[2]), C);
if C = {} then return 0 fi;
m:= infinity;
for t in C do
if abs(t[1]) < m then m:= abs(t[1]); r:= t[2];
elif abs(t[1]) = m and t[1] > 0 then r:= t[2]
fi
od;
r
end proc:
map(f, [$1..100]); # Robert Israel, Jul 21 2025
apply( {A384103(n)=for(x=1,n\/5, my(p=6*x+1, q=6*x-1, y=if((n-x)%p==0, (n-x)\p, (n+x)%p==0, -(n+x)\p, (n-x)%q==0, (n-x)\q, (n+x)%q==0,-(n+x)\q)); y && abs(y) <= x && return(y))}, [1..90])
a(1) = 6 - 4 = 2; a(2) = 11 - 8 = 3; a(3) = 13 - 9 = 4.
A046953 := proc(n) if n = 1 then 6 ; else for a from procname(n-1)+1 do if not isprime(6*a-1) then return a; end if; end do: end if; end proc: A046954 := proc(n) if n = 1 then 0 ; else for a from procname(n-1)+1 do if not isprime(6*a+1) then return a; end if; end do: end if; end proc: A173229 := proc(n) A046953(n)-A046954(n+1) ; end proc: seq(A173229(n),n=1..120) ; # R. J. Mathar, May 02 2010
a(1) = 6 + 4 = 10; a(2) = 11 + 8 = 19; a(3) = 13 + 9 = 22.
A046953:=Filtered([1..250], k-> not IsPrime(6*k-1));; A046954:=Filtered([0..250], n-> not IsPrime(6*n+1));; Print(List([1..80], j->A046953[j]+A046954[j+1])); # G. C. Greubel, Feb 21 2019
A046953:=[n: n in [1..250] | not IsPrime(6*n-1)]; A046954:=[n: n in [0..250] | not IsPrime(6*n+1)]; [A046953[n] + A046954[n+1]: n in [1..80]]; // G. C. Greubel, Feb 21 2019
A046953 := proc(n) if n = 1 then 6 ; else for a from procname(n-1)+1 do if not isprime(6*a-1) then return a; end if; end do: end if; end proc: A046954 := proc(n) if n = 1 then 0 ; else for a from procname(n-1)+1 do if not isprime(6*a+1) then return a; end if; end do: end if; end proc: A173231 := proc(n) A046953(n)+A046954(n+1) ; end proc: seq(A173231(n),n=1..120) ; # R. J. Mathar, May 02 2010
A046953:= Select[Range[250], !PrimeQ[6#-1] &]; A046954:= Select[Range[0, 250], !PrimeQ[6#+1] &]; Table[A046953[[n]] +A046954[[n+1]], {n,1,80}]
A046953=[n for n in (1..250) if not is_prime(6*n-1)]; A046954=[n for n in (0..250) if not is_prime(6*n+1)]; [A046953[n] + A046954[n+1] for n in (0..80)] # G. C. Greubel, Feb 21 2019
Select[Range[100], PrimeQ[6 # - 1] || PrimeQ[6 # + 1] &]
isok(k) = isprime(6*k-1) || isprime(6*k+1); \\ Michel Marcus, Oct 31 2024
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