A087679 -id:A087679 - cp's OEIS frontend

A377185 Average of twin prime pairs A014574 such that the largest odd divisor of the average is the midpoint of the cousin prime pairs A087679.

18, 30, 42, 60, 72, 138, 180, 198, 240, 312, 420, 462, 618, 660, 882, 1032, 1152, 1230, 1290, 1320, 1482, 1722, 1878, 2592, 2688, 2970, 3120, 3168, 3300, 3330, 3360, 3390, 3528, 3540, 3672, 4002, 4128, 4482, 5280, 5418, 5502, 6660, 6690, 6780, 7350, 7590, 7758, 8010, 8088, 8820, 9042

Offset: 1

Juri-Stepan Gerasimov, Oct 19 2024

nonn

			18 is in this sequence because:
(a) 18 is average of twin prime pair {17, 19};
(b) the largest odd divisor of 18 is 9;
(c) 9 is the midpoint of the cousin prime pair {7, 11}.

James C. McMahon, Table of n, a(n) for n = 1..10000

Cf. A000265, A014574, A087679.

[n: n in [4..10^4] | IsPrime(n-1) and IsPrime(n+1) and IsPrime((n div 2^Valuation(n, 2))-2) and IsPrime((n div 2^Valuation(n, 2))+2)];

lod[m_]:=Select[Divisors[m+1],OddQ][[-1]];Select[Prime[Range[1130]],PrimeQ[#+2]&&PrimeQ[lod[#]-2]&&PrimeQ[lod[#]+2]&]+1 (* James C. McMahon, Nov 17 2024 *)

A023200 Primes p such that p + 4 is also prime.

Original entry on oeis.org

3, 7, 13, 19, 37, 43, 67, 79, 97, 103, 109, 127, 163, 193, 223, 229, 277, 307, 313, 349, 379, 397, 439, 457, 463, 487, 499, 613, 643, 673, 739, 757, 769, 823, 853, 859, 877, 883, 907, 937, 967, 1009, 1087, 1093, 1213, 1279, 1297, 1303, 1423, 1429, 1447, 1483

Offset: 1

David W. Wilson

nonn

Smaller member p of cousin prime pairs (p, p+4).
A015913 contains the composite number 305635357, so it is different from both the present sequence and A029710. (305635357 is the only composite member of A015913 < 10^9.) - Jud McCranie, Jan 07 2001
Apart from the first term, all terms are of the form 6n + 1.
Complement of A067775 (primes p such that p + 4 is composite) with respect to A000040 (primes). With prime 2 also primes p such that q^2 + p is prime for some prime q (q = 3 if p = 2, q = 2 if p > 2). Subsequence of A232012. - Jaroslav Krizek, Nov 23 2013
Conjecture: The sequence is infinite and for every n, a(n+1) < a(n)^(1+1/n). Namely a(n)^(1/n) is a strictly decreasing function of n. - Jahangeer Kholdi and Farideh Firoozbakht, Nov 24 2014
From Alonso del Arte, Jan 12 2019: (Start)
If p splits in Z[sqrt(-2)], p + 4 is an inert prime in that domain. Likewise, if p splits in Z[sqrt(2)], p + 4 is an inert prime in that domain.
The only way for p or p + 4 to split in both domains is if it is congruent to 1 modulo 24, in which case the other prime is inert in both domains.
For example, 3 = (1 - sqrt(-2))*(1 + sqrt(-2)) but is inert in Z[sqrt(2)], while 7 = (3 - sqrt(2))*(3 + sqrt(2)) but is inert in Z[sqrt(-2)]. And also 11 = (3 - sqrt(-2))*(3 + sqrt(-2)) but 15 is composite in Z or any quadratic integer ring.
And 97 = (5 - 6*sqrt(-2))*(5 + 6*sqrt(-2)) = (1 - 7*sqrt(2))*(1 + 7*sqrt(2)), but 101 is inert in both Z[sqrt(-2)] and Z[sqrt(2)]. (End)

T. D. Noe, Table of n, a(n) for n = 1..10000
Andrew Granville and Greg Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
H. J. Weber, A Sieve for Cousin Primes, arXiv:1204.3795v1 [math.NT], 2012.
Eric Weisstein's World of Mathematics, Cousin Primes
Eric Weisstein's World of Mathematics, Twin Primes
Wikipedia, Cousin prime.
Index entries for primes, gaps between

Exactly the same as A029710 except for the exclusion of 3.

Cf. A000010, A003557, A007947, A046132, A098429, A000040, A010051.

a023200 n = a023200_list !! (n-1)
a023200_list = filter ((== 1) . a010051') $
               map (subtract 4) $ drop 2 a000040_list
-- Reinhard Zumkeller, Aug 01 2014

[p: p in PrimesUpTo(1500) | NextPrime(p)-p eq 4]; // Bruno Berselli, Apr 09 2013

A023200 := proc(n) option remember; if n = 1 then 3; else p := nextprime(procname(n-1)) ; while not isprime(p+4) do p := nextprime(p) ;  end do: p ; end if; end proc: # R. J. Mathar, Sep 03 2011

Select[Range[10^2], PrimeQ[#] && PrimeQ[# + 4] &] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
Select[Prime[Range[250]],PrimeQ[#+4]&] (* Harvey P. Dale, Oct 09 2023 *)

print1(3);p=7;forprime(q=11,1e3,if(q-p==4,print1(", "p)); p=q) \\ Charles R Greathouse IV, Mar 20 2013

a(n) = A046132(n) - 4 = A087679(n) - 2.

a(n) >> n log^2 n via the Selberg sieve. - Charles R Greathouse IV, Nov 20 2016

Definition modified by Vincenzo Librandi, Aug 02 2009

Definition revised by N. J. A. Sloane, Mar 05 2010

A046132 Larger member p+4 of cousin primes (p, p+4).

Original entry on oeis.org

7, 11, 17, 23, 41, 47, 71, 83, 101, 107, 113, 131, 167, 197, 227, 233, 281, 311, 317, 353, 383, 401, 443, 461, 467, 491, 503, 617, 647, 677, 743, 761, 773, 827, 857, 863, 881, 887, 911, 941, 971, 1013, 1091, 1097, 1217, 1283, 1301, 1307, 1427, 1433

Offset: 1

Eric W. Weisstein

nonn

A pair of cousin primes are primes of the form p and p+4 (where p+2 may or may not be a prime). - N. J. A. Sloane, Mar 18 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Cousin Primes
Wikipedia, Cousin prime.

Essentially the same as A031505. Cf. A023200, A029710, A098429.

Cf. A000040, A010051.

a046132 n = a046132_list !! (n-1)
a046132_list = filter ((== 1) . a010051') $ map (+ 4) a000040_list
-- Reinhard Zumkeller, Aug 01 2014

lst={};Do[p=Prime[n];If[PrimeQ[p4=p+4], (*Print[p4];*)AppendTo[lst, p4]], {n, 10^2}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)
Select[Prime[Range[300]],PrimeQ[#+4]&]+4 (* Harvey P. Dale, Dec 15 2017 *)

forprime(p=2,1e5,if(isprime(p-4),print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = A023200(n) + 4 = A087679(n) + 2.

a(n) = 3*A157834(n-1) + 2 = A029710(n-1) + 4 = 6*A056956(n-1) + 5 (thus a(n) mod 6 == 5), for all n>1. - M. F. Hasler, Jan 15 2013

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

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

A056956 Numbers n such that 6n+1 and 6n+5 are both primes.

Original entry on oeis.org

1, 2, 3, 6, 7, 11, 13, 16, 17, 18, 21, 27, 32, 37, 38, 46, 51, 52, 58, 63, 66, 73, 76, 77, 81, 83, 102, 107, 112, 123, 126, 128, 137, 142, 143, 146, 147, 151, 156, 161, 168, 181, 182, 202, 213, 216, 217, 237, 238, 241, 247, 248, 258, 261, 263, 266, 268, 277, 282

Offset: 1

Henry Bottomley, Jul 18 2000

nonn

Note that if prime p>3 then p mod 6 = 1 or 5.

			a(2)=2 since 6*2+1=13 and 6*2+5=17 are both prime.

M. F. Hasler, Table of n, a(n) for n = 1..5000

Cf. A002822, A024898, A024899, A059325.

Select[Range[300], And @@ PrimeQ /@ ({1, 5} + 6#) &] (* Ray Chandler, Jun 29 2008 *)

is(n)=isprime(n*6+1)&&isprime(n*6+5) \\ M. F. Hasler, Apr 05 2017

a(n) = (A023200(n+1)-1)/6 = (A046132(n+1)-5)/6 = A047847(n+1)/3

a(n) = floor(A087679(n+1)/6). - M. F. Hasler, Apr 05 2017

Edited by N. J. A. Sloane, Nov 07 2006

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 *)

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

A173037 Numbers k such that k-4, k-2, k+2 and k+4 are prime.

Original entry on oeis.org

9, 15, 105, 195, 825, 1485, 1875, 2085, 3255, 3465, 5655, 9435, 13005, 15645, 15735, 16065, 18045, 18915, 19425, 21015, 22275, 25305, 31725, 34845, 43785, 51345, 55335, 62985, 67215, 69495, 72225, 77265, 79695, 81045, 82725, 88815, 97845

Offset: 1

Juri-Stepan Gerasimov, Feb 07 2010

nonn

Average k of the four primes in two twin prime pairs (k-4, k-2) and (k+2, k+4) which are linked by the cousin prime pair (k-2, k+2).
All terms are odd composites; except for a(1) they are multiples of 5.
Subsequence of A087679, of A087680 and of A164385.
All terms except for a(1) are multiples of 15. - Zak Seidov, May 18 2014
One of (k-1, k, k+1) is always divisible by 7. - Fred Daniel Kline, Sep 24 2015
Terms other than a(1) must be equivalent to 1 mod 2, 0 mod 3, 0 mod 5, and 0,+/-1 mod 7. Taken together, this requires terms other than a(1) to have the form 210k+/-15 or 210k+105. However, not all numbers of that form belong to this sequence. - Keith Backman, Nov 09 2023

			9 is a term because 9-4 = 5 is prime, 9-2 = 7 is prime, 9+2 = 11 is prime and 9+4 = 13 is prime.

Klaus Brockhaus, Table of n, a(n) for n = 1..28388 (terms < 10^9).

Cf. A001359, A006512, A007530, A007811, A014561, A023200, A038800, A046132, A087680, A112540, A125855, A164385.

[ p+4: p in PrimesUpTo(100000) | IsPrime(p) and IsPrime(p+2) and IsPrime(p+6) and IsPrime(p+8) ]; // Klaus Brockhaus, Feb 09 2010

Select[Range[100000],AllTrue[#+{4,2,-2,-4},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 30 2015 *)

is(n)=isprime(n-4) && isprime(n-2) && isprime(n+2) && isprime(n+4) \\ Charles R Greathouse IV, Sep 24 2015

from sympy import primerange
def aupto(limit):
    p, q, r, alst = 2, 3, 5, []
    for s in primerange(7, limit+5):
        if p+2 == q and p+6 == r and p+8 == s: alst.append(p+4)
        p, q, r = q, r, s
    return alst
print(aupto(10**5)) # Michael S. Branicky, Feb 03 2022

For n >= 2, a(n) = 15*A112540(n-1). - Michel Marcus, May 19 2014
From Jeppe Stig Nielsen, Feb 18 2020: (Start)
For n >= 2, a(n) = 30*A014561(n-1) + 15.
For n >= 2, a(n) = 10*A007811(n-1) + 5.
a(n) = A007530(n) + 4.
a(n) = A125855(n) + 5. (End)

Edited and extended beyond a(9) by Klaus Brockhaus, Feb 09 2010

A088762 Numbers n such that (2n-1, 2n+3) is a cousin prime pair.

Original entry on oeis.org

2, 4, 7, 10, 19, 22, 34, 40, 49, 52, 55, 64, 82, 97, 112, 115, 139, 154, 157, 175, 190, 199, 220, 229, 232, 244, 250, 307, 322, 337, 370, 379, 385, 412, 427, 430, 439, 442, 454, 469, 484, 505, 544, 547, 607, 640, 649, 652, 712, 715, 724, 742, 745, 775, 784, 790

Offset: 1

Ray Chandler, Oct 26 2003

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..4200
Eric Weisstein's World of Mathematics, Cousin Primes

Essentially the same as A111981.

Cf. A087679, A088764, A088766, A088768, A088770.

[n: n in [1..1000] |IsPrime(2*n-1) and IsPrime(2*n+3)]; // Vincenzo Librandi, May 19 2017

Select[Range[2000],PrimeQ[2*#-1]&&PrimeQ[2*#+3]&] (* Vladimir Joseph Stephan Orlovsky, Dec 26 2010 *)

a(n) = (A087679(n)-1)/2 = (A023200(n)+1)/2 = (A046132(n)-3)/2.

cp's OEIS Frontend

A377185 Average of twin prime pairs A014574 such that the largest odd divisor of the average is the midpoint of the cousin prime pairs A087679.

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A023200 Primes p such that p + 4 is also prime.

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A046132 Larger member p+4 of cousin primes (p, p+4).

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

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

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Magma

Maple

Mathematica

PARI

A056956 Numbers n such that 6n+1 and 6n+5 are both primes.

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Mathematica

PARI

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

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