A087695 -id:A087695 - cp's OEIS frontend

A088763 a(n) = A087695(n)/2.

4, 5, 7, 8, 10, 13, 17, 20, 22, 25, 28, 32, 35, 38, 43, 50, 52, 53, 55, 67, 77, 80, 85, 88, 97, 98, 113, 115, 118, 127, 130, 133, 137, 140, 155, 157, 167, 175, 178, 185, 188, 193, 218, 223, 230, 232, 253, 272, 280, 283, 287, 295, 298, 302, 305, 308, 322, 325, 328, 340

Offset: 1

Ray Chandler, Oct 26 2003

nonn

A260689(a(n),1) = A264526(a(n)) = 3. - Reinhard Zumkeller, Nov 17 2015

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

Cf. A087695, A040040, A088765, A088767, A088769.

Cf. A260689, A264526.

a088763 = flip div 2 . a087695  -- Reinhard Zumkeller, Nov 17 2015

ZL:=[]:for p from 1 to 700 do if (isprime(p) and isprime(p+6) ) then ZL:=[op(ZL),(p+(p+6))/4]; fi; od; print(ZL); # Zerinvary Lajos, Mar 07 2007

f[n_]:=PrimeQ[n-3]&&PrimeQ[n+3]; lst={};Do[If[f[n],AppendTo[lst,n]],{n,2,8!,2}];lst/2 (* Vladimir Joseph Stephan Orlovsky, Oct 09 2009 *)

Offset corrected by Reinhard Zumkeller, Nov 17 2015

A023201 Primes p such that p + 6 is also prime. (Lesser of a pair of sexy primes.)

Original entry on oeis.org

5, 7, 11, 13, 17, 23, 31, 37, 41, 47, 53, 61, 67, 73, 83, 97, 101, 103, 107, 131, 151, 157, 167, 173, 191, 193, 223, 227, 233, 251, 257, 263, 271, 277, 307, 311, 331, 347, 353, 367, 373, 383, 433, 443, 457, 461, 503, 541, 557, 563, 571, 587, 593, 601, 607, 613, 641, 647

Offset: 1

David W. Wilson

T. D. Noe, Table of n, a(n) for n = 1..10000
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
Eric Weisstein's World of Mathematics, Sexy Primes. [The definition in this webpage is unsatisfactory, because it defines a "sexy prime" as a pair of primes.- _N. J. A. Sloane_, Mar 07 2021].
Wikipedia, Sexy prime.

A031924 (primes starting a gap of 6) and A007529 together give this (A023201).

Cf. A046117 (a(n)+6), A087695 (a(n)+3), A098428, A000040, A010051, A006489 (subsequence).

a023201 n = a023201_list !! (n-1)
a023201_list = filter ((== 1) . a010051 . (+ 6)) a000040_list
-- Reinhard Zumkeller, Feb 25 2013

[n: n in [0..40000] | IsPrime(n) and IsPrime(n+6)]; // Vincenzo Librandi, Aug 04 2010

A023201 := proc(n)
    option remember;
    if n = 1 then
        5;
    else
        for a from procname(n-1)+2 by 2 do
            if isprime(a) and isprime(a+6) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, May 28 2013

Select[Range[10^2], PrimeQ[ # ]&&PrimeQ[ #+6] &] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
Select[Prime[Range[120]],PrimeQ[#+6]&] (* Harvey P. Dale, Mar 20 2018 *)

is(n)=isprime(n+6)&&isprime(n) \\ Charles R Greathouse IV, Mar 20 2013

From M. F. Hasler, Jan 02 2020: (Start)
a(n) = A046117(n) - 6 = A087695(n) - 3.
A023201 = { p = A000040(k) | A000040(k+1) = p+6 or A000040(k+2) = p+6 } = A031924 U A007529. (End)

A046117 Primes p such that p-6 is also prime. (Upper of a pair of sexy primes.)

Original entry on oeis.org

11, 13, 17, 19, 23, 29, 37, 43, 47, 53, 59, 67, 73, 79, 89, 103, 107, 109, 113, 137, 157, 163, 173, 179, 197, 199, 229, 233, 239, 257, 263, 269, 277, 283, 313, 317, 337, 353, 359, 373, 379, 389, 439, 449, 463, 467, 509, 547, 563, 569, 577, 593, 599, 607, 613

Offset: 1

Eric W. Weisstein

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Eric Weisstein's World of Mathematics, Sexy Primes. [The definition in this webpage is unsatisfactory, because it defines a "sexy prime" as a pair of primes.- _N. J. A. Sloane_, Mar 07 2021].
Wikipedia, Sexy prime.

Cf. A023201, A098428, A087695.

[p:p in PrimesInInterval(7,650)| IsPrime(p-6)]; // Marius A. Burtea, Jan 03 2020

q=6; a={}; Do[p1=Prime[n]; p2=p1+q; If[PrimeQ[p2], AppendTo[a, p2]], {n, 7^2}]; a "and/or" Select[Prime[Range[3, 7^2]], PrimeQ[ # ]&&PrimeQ[ #-6]&] (* Vladimir Joseph Stephan Orlovsky, Aug 07 2008 *)
Select[Prime[Range[4,200]],PrimeQ[#-6]&] (* Harvey P. Dale, Mar 31 2018 *)

forprime(p=2,1e4,if(isprime(p-6),print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = A087695(n) + 3.

a(n) = A023201(n) + 6. - M. F. Hasler, Jan 02 2020

Name edited by M. F. Hasler, Jan 02 2020

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

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

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 *)

cp's OEIS Frontend

A088763 a(n) = A087695(n)/2.

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A023201 Primes p such that p + 6 is also prime. (Lesser of a pair of sexy primes.)

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A046117 Primes p such that p-6 is also prime. (Upper of a pair of sexy primes.)

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

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Magma

Maple

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

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