A087697 -id:A087697 - cp's OEIS frontend

A088767 a(n) = A087697(n)/2.

5, 6, 12, 15, 18, 27, 30, 33, 45, 48, 60, 72, 78, 87, 93, 102, 117, 132, 135, 138, 150, 162, 180, 183, 195, 213, 225, 228, 258, 282, 285, 297, 300, 303, 312, 327, 333, 342, 363, 375, 390, 402, 408, 423, 435, 480, 492, 495, 513, 528, 555, 558, 597, 612, 615, 642

Offset: 1

Ray Chandler, Oct 26 2003

nonn

Numbers n such that 2*n-7 [A089192] and 2*n+7 [A105760] are both prime. [Vincenzo Librandi, Jul 10 2010]

Cf. A087697, A040040, A088763, A088765, A088769.

Cf. A089192, A105760. [Vincenzo Librandi, Jan 18 2009]

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

A087679 Numbers k such that both k+2 and k-2 are prime.

Original entry on oeis.org

5, 9, 15, 21, 39, 45, 69, 81, 99, 105, 111, 129, 165, 195, 225, 231, 279, 309, 315, 351, 381, 399, 441, 459, 465, 489, 501, 615, 645, 675, 741, 759, 771, 825, 855, 861, 879, 885, 909, 939, 969, 1011, 1089, 1095, 1215, 1281, 1299, 1305, 1425, 1431, 1449, 1485

Offset: 1

Zak Seidov, Sep 27 2003

Essentially the same as A029708: a(n) = A029708(n-1) for n>=2.
Midpoint of cousin prime pairs.
The only prime is 5. All other terms are multiples of 3. - Zak Seidov, May 19 2014

M. F. Hasler, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Cousin Primes

Cf. A014574, A087678-A087683, A087695-A087697, A088762.

ZL:=[]:for p from 1 to 1485 do if (isprime(p) and isprime(p+4) ) then ZL:=[op(ZL),(p+(p+4))/2]; fi; od; print(ZL); # Zerinvary Lajos, Mar 07 2007

lst={};Do[If[PrimeQ[n-2]&&PrimeQ[n+2],AppendTo[lst,n]],{n,3,8!,2}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 14 2009 *)

s=[]; for(n=1, 2000, if(isprime(n-2) && isprime(n+2), s=concat(s, n))); s \\ Colin Barker, May 19 2014

is_A087679(n)={isprime(n-2) && isprime(n+2)} \\ For numbers >> 10^12 one should add conditions {n%6==3 && ... || n==5} or consider only such numbers congruent to 3 (mod 6). - M. F. Hasler, Apr 05 2017

a(n) = (A023200(n) + A046132(n))/2 = A023200(n) + 2 = A046132(n) - 2.

a(n+1) = A056956(n)*6 + 3 = A157834(n)*3; a(n) = A088762(n)*2 + 1. - M. F. Hasler, Apr 05 2017

More terms from Ray Chandler, Oct 26 2003

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.

A087678 Numbers n such that n + 9 and n - 9 are both prime.

Original entry on oeis.org

14, 20, 22, 28, 32, 38, 50, 52, 62, 70, 80, 88, 92, 98, 118, 122, 140, 148, 158, 172, 182, 188, 190, 202, 220, 232, 242, 248, 260, 272, 302, 322, 340, 358, 388, 392, 410, 430, 440, 448, 452, 458, 470, 500, 512, 532, 578, 608, 610, 622, 650, 652, 668, 682, 692

Offset: 1

Zak Seidov, Sep 27 2003

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A014574, A087679-A087683, A087695-A087697, A088769.

f[n_]:=PrimeQ[n-9]&&PrimeQ[n+9]; lst={};Do[If[f[n],AppendTo[lst,n]],{n,6,8!,2}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 09 2009 *)
Select[Range[10,700],AllTrue[#+{9,-9},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 23 2019 *)

A087683 Numbers n such that n + 10 and n - 10 are both prime.

Original entry on oeis.org

13, 21, 27, 33, 51, 57, 63, 69, 93, 99, 117, 141, 147, 183, 189, 201, 261, 267, 273, 303, 321, 327, 357, 363, 369, 399, 411, 429, 453, 477, 489, 513, 531, 567, 597, 603, 609, 651, 663, 729, 819, 849, 867, 873, 897, 957, 981, 987

Offset: 1

Zak Seidov, Sep 27 2003

3 divides every term except the first. - T. D. Noe, May 14 2008

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A014574, A087678-A087682, A087695-A087697, A088770.

f[n_]:=PrimeQ[n-10]&&PrimeQ[n+10]; lst={};Do[If[f[n],AppendTo[lst,n]],{n,9,8!,2}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 09 2009 *)
Select[Range[10,1000],AllTrue[#+{10,-10},PrimeQ]&] (* Harvey P. Dale, Jul 29 2024 *)

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

A087681 Numbers n such that n + 6 and n - 6 are both prime.

Original entry on oeis.org

11, 13, 17, 23, 25, 35, 37, 47, 53, 65, 67, 73, 77, 95, 103, 107, 133, 143, 145, 157, 173, 185, 187, 205, 217, 233, 235, 245, 257, 263, 275, 277, 287, 343, 353, 373, 395, 403, 415, 425, 427, 437, 455, 473, 485, 493, 497, 515, 563, 593, 607, 613, 625, 637, 647

Offset: 1

Zak Seidov, Sep 27 2003

Many terms are of the form 5 + n + n^2 or 5 + 2*n^2: A054794.

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

Cf. A054794.

Cf. A014574, A087678-A087683, A087695-A087697, A088766.

f[n_]:=PrimeQ[n-6]&&PrimeQ[n+6]; lst={}; Do[If[f[n],AppendTo[lst,n]],{n,2,7!,1}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)
Select[Range[2,700],AllTrue[#+{6,-6},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 26 2019 *)

A087696 Numbers n such that n + 5 and n - 5 are both prime.

Original entry on oeis.org

8, 12, 18, 24, 36, 42, 48, 66, 78, 84, 102, 108, 132, 144, 162, 168, 186, 228, 234, 246, 276, 288, 312, 342, 354, 378, 384, 414, 426, 438, 444, 462, 504, 552, 582, 612, 636, 648, 678, 696, 714, 738, 756, 792, 816, 834, 858, 882, 924, 942, 972

Offset: 1

Zak Seidov, Sep 27 2003

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A014574, A087678-A087683, A087695-A087697, A088765.

Select[Range[5,1000],AllTrue[#+{5,-5},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 08 2014 *)

isok(n) = isprime(n-5) && isprime(n+5); \\ Michel Marcus, Sep 02 2019

A087680 Numbers n such that n + 4 and n - 4 are both prime.

Original entry on oeis.org

7, 9, 15, 27, 33, 57, 63, 75, 93, 105, 135, 153, 177, 195, 237, 267, 273, 363, 393, 405, 435, 453, 483, 495, 567, 573, 597, 603, 657, 687, 705, 723, 747, 765, 825, 915, 933, 987, 1017, 1035, 1065, 1113, 1167, 1197, 1227, 1233, 1287, 1293, 1323, 1377, 1443

Offset: 1

Zak Seidov, Sep 27 2003

All terms > 7 (prime) are divisible by 3. Also note that n-4 and n+4 are not necessarily consecutive primes. First case when n-4 and n+4 are consecutive primes is for n=93 with n-4=89 and n+4=97. - Zak Seidov, Apr 22 2015

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A023202, A014574, A087678-A087683, A087695-A087697, A088764.

ZL:=[]:for p from 1 to 1444 do if (isprime(p) and isprime(p+8) ) then ZL:=[op(ZL),(p+(p+8))/2]; fi; od; print(ZL); # Zerinvary Lajos, Mar 07 2007

f[n_]:=PrimeQ[n-4]&&PrimeQ[n+4]; lst={};Do[If[f[n],AppendTo[lst,n]],{n,3,8!,2}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 09 2009 *)
Select[Prime[Range[250]],PrimeQ[#+8]&]+4 (* Harvey P. Dale, May 21 2023 *)

a(n) = A023202(n) + 4. - Michel Marcus, Apr 22 2015

More terms from Ray Chandler, Oct 26 2003

cp's OEIS Frontend

A088767 a(n) = A087697(n)/2.

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

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A087679 Numbers k such that both k+2 and k-2 are prime.

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Maple

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

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Haskell

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A087678 Numbers n such that n + 9 and n - 9 are both prime.

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Mathematica

A087683 Numbers n such that n + 10 and n - 10 are both prime.

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Mathematica

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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