A087696 -id:A087696 - cp's OEIS frontend

A088765 a(n) = A087696(n)/2.

4, 6, 9, 12, 18, 21, 24, 33, 39, 42, 51, 54, 66, 72, 81, 84, 93, 114, 117, 123, 138, 144, 156, 171, 177, 189, 192, 207, 213, 219, 222, 231, 252, 276, 291, 306, 318, 324, 339, 348, 357, 369, 378, 396, 408, 417, 429, 441, 462, 471, 486, 507, 513, 522, 528, 546

Offset: 1

Ray Chandler, Oct 26 2003

nonn

"Example 3: Ordinary twins 2a +- 1 for a = 2, 3, 6, 9, 15, . . . have D = 1 and are in class I. For D = 3, the twins 2a +- 3 occur for a = 4, 5, 7, 8, 10"; the latter is this sequence, from p. 3 of Weber. - Jonathan Vos Post, Feb 14 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..3100
H. J. Weber, Exceptional Prime Number Twins, Triplets and Multiplets, arXiv:1102.3075 [math.NT], 2011.

Cf. A087696, A040040, A088763, A088767, A088769.

[n/2: n in [3..2000] |IsPrime(n+5) and IsPrime(n-5)]; // Vincenzo Librandi, May 20 2017

Select[Range[3, 2000], PrimeQ[# + 5] && PrimeQ[# - 5] &] / 2 (* Vincenzo Librandi, May 20 2017 *)

Offset changed from 0 to 1 by Vincenzo Librandi, May 21 2017

A082467 Least k>0 such that n-k and n+k are both primes.

Original entry on oeis.org

1, 2, 1, 4, 3, 2, 3, 6, 1, 6, 3, 2, 3, 6, 1, 12, 3, 2, 9, 6, 5, 6, 3, 4, 9, 12, 1, 12, 9, 4, 3, 6, 5, 6, 9, 2, 3, 12, 1, 24, 3, 2, 15, 6, 5, 12, 3, 8, 9, 6, 7, 12, 3, 4, 15, 12, 1, 18, 9, 4, 3, 6, 5, 6, 15, 2, 3, 12, 1, 6, 15, 4, 3, 6, 5, 18, 9, 2, 15, 24, 5, 12, 3, 14, 9, 18, 7, 12, 9, 4, 15, 6, 7, 30, 9

Offset: 4

Benoit Cloitre, Apr 27 2003

nonn

The existence of k>0 for all n >= 4 is equivalent to the strong Goldbach Conjecture that every even number >= 8 is the sum of two distinct primes.
n and k are coprime, because otherwise n + k would be composite. So the rational sequence r(n) = a(n)/n = k/n is injective. - Jason Kimberley, Sep 21 2011
Because there are arbitrarily many composites from m!+2 to m!+m, there are also arbitrarily large a(n) but they increase very slowly. The twin prime conjecture implies that infinitely many a(n) are 1. - Juhani Heino, Apr 09 2020

			n=10: k=3 because 10-3 and 10+3 are both prime and 3 is the smallest k such that n +/- k are both prime.

Klaus Brockhaus, Table of n, a(n) for n = 4..5000
OEIS (Plot 2), log_10(A082467(n)/n) vs n
J. S. Kimberley, A082467

Cf. A087695, A087696, A087697, A087678, A087679, A087680, A087681, A087682, A087683, A087711.
Cf. A129301 (records), A129302 (where records occur).
Cf. A047160 (allows k=0).
Cf. A078611 (subset for prime n).

A082467 := func; [A082467(n):n in [4..98]]; // Jason Kimberley, Sep 03 2011

A082467 := proc(n) local k; k := 1+irem(n,2);
while n > k do if isprime(n-k) then if isprime(n+k)
then RETURN(k) fi fi; k := k+2 od; print("Goldbach erred!") end:
seq(A082467(i),i=4..90); # Peter Luschny, Sep 21 2011

f[n_] := Block[{k}, If[OddQ[n], k = 2, k = 1]; While[ !PrimeQ[n - k] || !PrimeQ[n + k], k += 2]; k]; Table[ f[n], {n, 4, 98}] (* Robert G. Wilson v, Mar 28 2005 *)

a(n)=if(n<0,0,k=1; while(isprime(n-k)*isprime(n+k) == 0,k++); k)

A078496(n)-a(n) = A078587(n)+a(n) = n.

Entries checked by Klaus Brockhaus, Apr 08 2007

A087695 Numbers n such that n + 3 and n - 3 are both prime.

Original entry on oeis.org

8, 10, 14, 16, 20, 26, 34, 40, 44, 50, 56, 64, 70, 76, 86, 100, 104, 106, 110, 134, 154, 160, 170, 176, 194, 196, 226, 230, 236, 254, 260, 266, 274, 280, 310, 314, 334, 350, 356, 370, 376, 386, 436, 446, 460, 464, 506, 544, 560, 566, 574, 590, 596

Offset: 1

Zak Seidov, Sep 27 2003

A010051(a(n)-3) * A010051(a(n)+3) = 1. - Reinhard Zumkeller, Nov 17 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A014574, A087678-A087683, A087696, A087697, A088763, A046117, A010051.

a087695 n = a087695_list !! (n-1)
a087695_list = filter
   (\x -> a010051' (x - 3) == 1 && a010051' (x + 3) == 1) [2, 4 ..]
-- Reinhard Zumkeller, Nov 17 2015

ZL:=[]:for p from 1 to 600 do if (isprime(p) and isprime(p+6) ) then ZL:=[op(ZL),(p+(p+6))/2]; fi; od; print(ZL); # Zerinvary Lajos, Mar 07 2007

lst={};Do[If[PrimeQ[n-3]&&PrimeQ[n+3], AppendTo[lst, n]], {n, 10^3}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 08 2008 *)
Select[Range[600],AllTrue[#+{3,-3},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 06 2015 *)

p=2; q=3; forprime(r=5,1e3, if(q-p<7 && (q-p==6 || r-p==6), print1(p+3", ")); p=q; q=r) \\ Charles R Greathouse IV, May 22 2018

a(n) = A046117(n) - 3.

A087697 Numbers k such that k + 7 and k - 7 are both prime.

Original entry on oeis.org

10, 12, 24, 30, 36, 54, 60, 66, 90, 96, 120, 144, 156, 174, 186, 204, 234, 264, 270, 276, 300, 324, 360, 366, 390, 426, 450, 456, 516, 564, 570, 594, 600, 606, 624, 654, 666, 684, 726, 750, 780, 804, 816, 846, 870, 960, 984, 990

Offset: 1

Zak Seidov, Sep 27 2003

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A014574, A087678, A087679, A087680, A087681, A087682, A087683, A087695, A087696, A088767.

[n: n in [5..1000] | IsPrime(n-7) and IsPrime(n+7)]; // Vincenzo Librandi, Jul 23 2018

select(t -> isprime(t+7) and isprime(t-7), [seq(i,i=8..1000,2)]); # Robert Israel, Jul 22 2018

Rest[Select[Range[1000], PrimeQ[# - 7] && PrimeQ[# + 7] &]] (* Vincenzo Librandi, Jul 23 2018 *)

isok(n) = isprime(n-7) && isprime(n+7); \\ Michel Marcus, Jul 23 2018

A087711 a(n) = smallest number k such that both k-n and k+n are primes.

Original entry on oeis.org

2, 4, 5, 8, 7, 8, 11, 10, 11, 14, 13, 18, 17, 16, 17, 22, 21, 20, 23, 22, 23, 26, 25, 30, 29, 28, 33, 32, 31, 32, 37, 36, 35, 38, 37, 38, 43, 42, 41, 44, 43, 48, 47, 46, 57, 52, 51, 50, 53, 52, 53, 56, 55, 56, 59, 58, 75, 70, 69, 72, 67, 66, 65, 68, 67, 72, 71, 70, 71, 80, 81, 78

Offset: 0

Zak Seidov, Sep 28 2003

Let b(n), c(n) and d(n) be respectively, smallest number m such that phi(m-n) + sigma(m+n) = 2n, smallest number m such that phi(m+n) + sigma(m-n) = 2n and smallest number m such that phi(m-n) + sigma(m+n) = phi(m+n) + sigma(m-n), we conjecture that for each positive integer n, a(n)=b(n)=c(n)=d(n). Namely we conjecture that for each positive integer n, a(n) < A244446(n), a(n) < A244447(n) and a(n) < A244448(n). - Jahangeer Kholdi and Farideh Firoozbakht, Sep 05 2014

			n=10: k=13 because 13-10 and 13+10 are both prime and 13 is the smallest k such that k +/- 10 are both prime
4-1=3, prime, 4+1=5, prime; 5-2=3, 5+2=7; 8-3=5, 8+3=11; 9-4=5, 9+4=13, ...

Zak Seidov, Table of n, a(n) for n = 0..1000

Cf. A087695, A087696, A087697, A087678, A087679, A087680, A087681, A087682, A087683.
Cf. A082467. See A137169 for another version.
Cf. A244446, A244447, A244448.
Cf. A020483.

distance:=function(n); k:=n+2; while not IsPrime(k-n) or not IsPrime(k+n) do k:=k+1; end while; return k; end function; [ distance(n): n in [1..71] ]; /* Klaus Brockhaus, Apr 08 2007 */

Primes:= select(isprime,{seq(2*i+1,i=1..10^3)}):
a[0]:= 2:
for n from 1 do
  Q:= Primes intersect map(t -> t-2*n,Primes);
  if nops(Q) = 0 then break fi;
  a[n]:= min(Q) + n;
od:
seq(a[i],i=0..n-1); # Robert Israel, Sep 08 2014

s = ""; k = 0; For[i = 3, i < 22^2, If[PrimeQ[i - k] && PrimeQ[i + k], s = s <> ToString[i] <> ","; k++ ]; i++ ]; Print[s] (* Vladimir Joseph Stephan Orlovsky, Apr 03 2008 *)
snk[n_]:=Module[{k=n+1},While[!PrimeQ[k+n]||!PrimeQ[k-n],k++];k]; Array[ snk,80,0] (* Harvey P. Dale, Dec 13 2020 *)

a(n)=my(k);while(!isprime(k-n) || !isprime(k+n),k++);return(k) \\ Edward Jiang, Sep 05 2014

a(n) = A020483(n)+n for n >= 1. - Robert Israel, Sep 08 2014

Entries checked by Klaus Brockhaus, Apr 08 2007

A154713 Cubes such that cube-+5 are primes.

Original entry on oeis.org

8, 1728, 110592, 287496, 474552, 2000376, 7077888, 34012224, 191102976, 401947272, 631628712, 5890514616, 14996130696, 15550119936, 19421724672, 32339798856, 35158608576, 62949797352, 68518346688, 76657300992

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 14 2009

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A144938, A154709, A154710, A154711, A154712

lst={};Do[p=n^3;If[PrimeQ[p-5]&&PrimeQ[p+5],AppendTo[lst,p]],{n,2,2*7!,2}];lst
Select[Range[4300]^3,And@@PrimeQ[#+{5,-5}]&] (* Harvey P. Dale, Jun 19 2012 *)

A087696 INTERSECT A000578. [From R. J. Mathar, Jan 15 2009]

A293271 Numbers n such that n - p and n + p are both prime for some prime p.

Original entry on oeis.org

5, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 30, 32, 34, 36, 39, 40, 42, 44, 45, 46, 48, 50, 54, 56, 60, 64, 66, 69, 70, 72, 76, 78, 81, 84, 86, 90, 92, 96, 99, 100, 102, 104, 105, 106, 108, 110, 111, 114, 116, 120, 126, 129, 130, 132, 134, 138, 140, 142

Offset: 1

Gionata Neri, Oct 04 2017

nonn

Apart from a(1), all terms are composite.

Union of A087679 and 2*A063713. - Robert Israel, Oct 09 2017

Cf. A087679, A087695, A087696, A087697 (subsequences).

Cf. A063713.

filter:= proc(n) local k;
  k:= 1;
  while k < n do
    k:= nextprime(k);
    if isprime(n+k) and isprime(n-k) then return true fi
  od;
  false
end proc:
select(filter, [$1..1000]); # Robert Israel, Oct 09 2017

Select[Range@ 142, Function[n, AnyTrue[Prime@ Range@ PrimePi@ n, PrimeQ[n + {-#, #}] == {True, True} &]]] (* Michael De Vlieger, Oct 09 2017 *)

a(n) = forprime(p=1, n, i=n-p; j=n+p; if(isprime(i)&&isprime(j), n; break))

cp's OEIS Frontend

A088765 a(n) = A087696(n)/2.

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A082467 Least k>0 such that n-k and n+k are both primes.

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A087695 Numbers n such that n + 3 and n - 3 are both prime.

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A087697 Numbers k such that k + 7 and k - 7 are both prime.

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A087711 a(n) = smallest number k such that both k-n and k+n are primes.

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Magma

Maple

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A154713 Cubes such that cube-+5 are primes.

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Mathematica

Formula

A293271 Numbers n such that n - p and n + p are both prime for some prime p.

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