A046132 -id:A046132 - cp's OEIS frontend

A186243 Numbers k such that 6k-5 and 6k-1 are both primes.

2, 3, 4, 7, 8, 12, 14, 17, 18, 19, 22, 28, 33, 38, 39, 47, 52, 53, 59, 64, 67, 74, 77, 78, 82, 84, 103, 108, 113, 124, 127, 129, 138, 143, 144, 147, 148, 152, 157, 162, 169, 182, 183, 203, 214, 217, 218, 238, 239, 242, 248, 249, 259, 262, 264, 267, 269

Offset: 1

Jonathan Vos Post, Feb 15 2011

Numbers k such that 6*k-5 and 6*k-1 are cousin primes. The D = 2 numbers in class II, from page 3 of Weber. - Jonathan Vos Post, Feb 14 2011

			a(3) = 4 because 6*4-5 = 19 is prime, and 6*4-1 = 23 is prime.

Ivan Neretin, Table of n, a(n) for n = 1..10000
H. J. Weber, Exceptional Prime Number Twins, Triplets and Multiplets, arXiv:1102.3075 [math.NT], 2011.
Eric W. Weisstein, Cousin Primes.

Cf. A023200, A046132, A088765.

Select[Range[400], PrimeQ[6#-5] && PrimeQ[6#-1] &] (* Alonso del Arte, Feb 16 2011 *)

{k such that 6*k-5 is in A023200} = {k such that 6*k-1 is in A046132}.

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.

A092146 Primes of the form p + 10 where p is a prime.

Original entry on oeis.org

13, 17, 23, 29, 41, 47, 53, 71, 83, 89, 107, 113, 137, 149, 167, 173, 191, 233, 239, 251, 281, 293, 317, 347, 359, 383, 389, 419, 431, 443, 449, 467, 509, 557, 587, 617, 641, 653, 683, 701, 719, 743, 761, 797, 821, 839, 863, 887, 929, 947, 977, 1019, 1031

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Mar 31 2004

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A006512, A023203, A046132, A046117, A092402.

lst={};Do[p=Prime[n];If[PrimeQ[p=p+10],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 29 2009 *)
Select[Prime[Range[200]]+10,PrimeQ] (* Harvey P. Dale, Aug 03 2018 *)

a(n) = 10 + A023203(n). - Alois P. Heinz, Feb 27 2020

A228917 Number of undirected circular permutations i_0, i_1, ..., i_n of 0, 1, ..., n such that i_0+i_1, i_1+i_2, ...,i_{n-1}+i_n, i_n+i_0 are among those k with 6k-1 and 6k+1 twin primes.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 5, 2, 12, 39, 98, 526, 2117, 6663, 15043, 68403, 791581, 4826577, 19592777, 102551299, 739788968, 4449585790, 36547266589, 324446266072, 2743681178070

Offset: 1

Zhi-Wei Sun, Sep 08 2013

Conjecture: a(n) > 0 for all n > 0.
This implies the twin prime conjecture, and it is similar to the prime circle problem mentioned in A051252.
For each n = 2,3,... construct an undirected simple graph T(n) with vertices 0,1,...,n which has an edge connecting two distinct vertices i and j if and only if 6*(i+j)-1 and 6*(i+j)+1 are twin primes. Then a(n) is just the number of Hamiltonian cycles contained in T(n). Thus a(n) > 0 if and only if T(n) is a Hamilton graph.
Zhi-Wei Sun also made the following similar conjectures for odd primes, Sophie Germain primes, cousin primes and sexy primes:
(1) For any integer n > 0, there is a permutation i_0, i_1, ..., i_n of 0, 1, ..., n such that i_0+i_1, i_1+i_2, ..., i_{n-1}+i_n, i_n+i_0 are integers of the form (p-1)/2, where p is an odd prime. Also, we may replace the above (p-1)/2 by (p+1)/4 or (p-1)/6; when n > 4 we may substitute (p-1)/4 for (p-1)/2.
(2) For any integer n > 2, there is a permutation i_0, i_1, ..., i_n of 0, 1,..., n  such that i_0+i_1, i_1+i_2, ..., i_{n-1}+i_n, i_n+i_0 are integers of the form (p+1)/6, where p is a Sophie Germain prime.
(3) For any integer n > 3, there is a permutation i_0, i_1, ..., i_n of 0, 1,..., n  such that i_0+i_1, i_1+i_2, ..., i_{n-1}+i_n, i_n+i_0 are among those integers k with 6*k+1 and 6*k+5 both prime.
(4) For any integer n > 4, there is a permutation i_0, i_1, ..., i_n of 0, 1,..., n  such that i_0+i_1, i_1+i_2, ..., i_{n-1}+i_n, i_n+i_0 are among those integers k with 2*k-3 and 2*k+3 both prime.

			a(n) = 1 for n = 1,2,3 due to the permutation (0,...,n).
a(4) = 2 due to the permutations (0,1,4,3,2) and (0,2,1,4,3).
a(5) = 2 due to the permutations (0,1,4,3,2,5), (0,3,4,1,2,5).
a(6) = 2 due to the permutations
  (0,1,6,4,3,2,5) and (0,3,4,6,1,2,5).
a(7) = 5 due to the permutations
  (0,1,6,4,3,2,5,7), (0,1,6,4,3,7,5,2), (0,2,1,6,4,3,7,5),
  (0,3,4,6,1,2,5,7), (0,5,2,1,6,4,3,7).
a(8) = 2 due to the permutations
  (0,1,6,4,8,2,3,7,5) and (0,1,6,4,8,2,5,7,3).
a(9) = 12 due to the permutations
  (0,1,6,4,3,9,8,2,5,7), (0,1,6,4,8,9,3,2,5,7),
  (0,1,6,4,8,9,3,7,5,2), (0,2,1,6,4,8,9,3,7,5),
  (0,2,8,9,1,6,4,3,7,5), (0,3,4,6,1,9,8,2,5,7),
  (0,3,9,1,6,4,8,2,5,7), (0,3,9,8,4,6,1,2,5,7),
  (0,5,2,1,6,4,8,9,3,7), (0,5,2,8,4,6,1,9,3,7),
  (0,5,2,8,9,1,6,4,3,7), (0,5,7,3,9,1,6,4,8,2).
a(10) > 0 due to the permutation (0,5,2,3,9,1,6,4,8,10,7).
a(11) > 0 due to the permutation (0,10,8,9,3,7,11,6,4,1,2,5).
a(12) > 0 due to the permutation
        (0, 5, 2, 1, 6, 4, 3, 9, 8, 10, 7, 11, 12).

Zhi-Wei Sun, Twin primes and circular permutations, a message to Number Theory List, Sept. 8, 2013.
Z.-W. Sun, Some new problems in additive combinatorics, arXiv preprint arXiv:1309.1679 [math.NT], 2013-2014.

Cf. A000040, A001359, A006512, A023200, A046132, A023201, A046117, A005384, A051252, A228766, A228860, A228886.

(* A program to compute required circular permutations for n = 7. To get "undirected" circular permutations, we should identify a circular permutation with the one of the opposite direction; for example, (0,7,5,2,3,4,6,1) is identical to (0,1,6,4,3,2,5,7) if we ignore direction. Thus a(7) is half of the number of circular permutations yielded by this program. *)
tp[n_]:=tp[n]=PrimeQ[6n-1]&&PrimeQ[6n+1]
V[i_]:=Part[Permutations[{1,2,3,4,5,6,7}],i]
m=0
Do[Do[If[tp[If[j==0,0,Part[V[i],j]]+If[j<7,Part[V[i],j+1],0]]==False,Goto[aa]],{j,0,7}];
m=m+1;Print[m,":"," ",0," ",Part[V[i],1]," ",Part[V[i],2]," ",Part[V[i],3]," ",Part[V[i],4]," ",Part[V[i],5]," ",Part[V[i],6]," ",Part[V[i],7]];Label[aa];Continue,{i,1,7!}]

a(10)-a(25) from Max Alekseyev, Sep 12 2013

A157834 Numbers n such that 3n-2 and 3n+2 are both prime.

Original entry on oeis.org

3, 5, 7, 13, 15, 23, 27, 33, 35, 37, 43, 55, 65, 75, 77, 93, 103, 105, 117, 127, 133, 147, 153, 155, 163, 167, 205, 215, 225, 247, 253, 257, 275, 285, 287, 293, 295, 303, 313, 323, 337, 363, 365, 405, 427, 433, 435, 475, 477, 483, 495, 497, 517

Offset: 1

Kyle D. Balliet, Mar 07 2009

nonn

Barycenter of cousin primes (A029708; see also A029710, A023200, A046132), divided by 3. When p>3 and p+4 both are prime, then p = 1 (mod 6) and p+2 = 3 (mod 6). - M. F. Hasler, Jan 14 2013

			15*3 +/- 2 = 43,47 (both prime).

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

Intersection of A024893 and A153183.

[n: n in [1..1000]|IsPrime(3*n-2)and IsPrime(3*n+2)] // Vincenzo Librandi, Dec 13 2010

select(t -> isprime(3*t+2) and isprime(3*t-2), [seq(t,t=3..1000,2)]); # Robert Israel, May 28 2017

Select[Range[600],AllTrue[3#+{2,-2},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 03 2019 *)

Intersection of A024893 and A153183.
a(n) = A029708(n)/3. - Zak Seidov, Aug 07 2009
a(n) = A056956(n)*2+1 = (A029710(n)+2)/3 = (A023200(n+1)+2)/3. - M. F. Hasler, Jan 14 2013

A176130 Lesser of a pair (p,p+4) of cousin primes whose arithmetic mean p+2 is a square number.

Original entry on oeis.org

7, 79, 223, 439, 1087, 13687, 56167, 74527, 91807, 95479, 149767, 184039, 194479, 199807, 263167, 314719, 328327, 370879, 651247, 804607, 1071223, 1147039, 1238767, 1306447, 1520287, 1535119, 1718719, 2442967, 2595319, 2614687

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 09 2010

nonn

Necessarily p = 9 * (2*m - 1)^2 - 2.

			(7 + 11)/2 = 3^2, 1st term is prime(4) = 7.
(79 + 83)/2 = 9^2, 2nd term is prime(22) = 79.
m = 173 = prime(40): 21st term is p = 1071223 = prime(83637), p+2 = 3^4 * 5^2 * 23^2.
60th term is p = 27029599 = prime(1684797): p+2 = 3^2 * 1733^2.

L. E. Dickson, History of the Theory of numbers, vol. 2: Diophantine Analysis, Dover Publications 2005.
H. Pieper, Zahlen aus Primzahlen. Eine Einfuehrung in die Zahlentheorie. VEB Deutscher Verlag der Wissenschaften, 2. Aufl., 1984.
A. Warusfel, Les nombres et leurs mystères, Edition du Seuil, Paris 1980.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A023200, A046132, A174454.

Select[Range[1617]^2 - 2, And @@ PrimeQ[# + {0, 4}] &] (* Amiram Eldar, Dec 24 2019 *)

isok(n) = isprime(n) && isprime(n+4) && issquare(n+2) \\ Michel Marcus, Jul 22 2013

forstep(n=3,1e4,2,if(isprime(n^2-2)&&isprime(n^2+2),print1(n^2-2", "))) \\ Charles R Greathouse IV, Jul 23 2013

Edited by D. S. McNeil, Nov 18 2010

A178228 Numbers k such that (k^3 - 2, k^3 + 2) is a pair of cousin primes (see A178227).

Original entry on oeis.org

129, 189, 369, 435, 549, 555, 561, 819, 1245, 1491, 1719, 1779, 1839, 1875, 1935, 2175, 2289, 2415, 2451, 2595, 2709, 2769, 3141, 3441, 4401, 4611, 4851, 5655, 5775, 6075, 6099, 6795, 6969, 7125, 7239, 7365, 8109, 8139, 8325, 8361, 8385, 8535, 8685, 9591, 9765

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 23 2010

nonn

Necessarily k is an odd multiple of 3, Least significant digit of k is e = 1, 5 or 9 (3^3 - 2, 7^3 + 2 are multiples of 5).

			189 is a term since 189^3 - 2 = 6751267 = prime(460792), 189^3 + 2 = 6751271 = prime(460793).
12471 is a term since 12471^3 - 2 = 1939562763109 = prime(i), i = 71166976775, 12471^3 + 2 = 1939562763113 = prime(i+1).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A023200, A046132, A090121, A164834, A172494, A174370, A176130, A176229, A178227.

Select[Range[10^4], And @@ PrimeQ[#^3 + {-2, 2}] &] (* Amiram Eldar, Dec 24 2019 *)

for(n=1,10000,my(p1=n^3-2,p2=n^3+2);if(isprime(p1)&&isprime(p2)&&ispower((p1+p2)/2,3),print1(n,", "))) \\ Hugo Pfoertner, Dec 24 2019

Edited by N. J. A. Sloane, May 23 2010

a(1) and a(21) inserted by Amiram Eldar, Dec 24 2019

A187757 Number of ways to write n=x+y (x,y>0) with 6x-1, 6x+1, 6y+1 and 6y+5 all prime.

Original entry on oeis.org

0, 1, 2, 3, 2, 2, 2, 4, 3, 2, 2, 3, 4, 4, 2, 3, 2, 6, 6, 5, 4, 2, 6, 5, 4, 4, 2, 6, 4, 4, 4, 3, 5, 7, 5, 5, 3, 4, 9, 5, 6, 4, 5, 6, 4, 5, 5, 6, 7, 6, 6, 3, 7, 7, 6, 6, 4, 6, 6, 5, 6, 4, 7, 6, 7, 2, 3, 7, 7, 7, 5, 3, 5, 5, 7, 8, 5, 8, 8, 4, 5, 4, 10, 10, 6, 6, 2, 9, 6, 9, 7, 1, 8, 4, 5, 7, 3, 9, 5, 3

Offset: 1

Zhi-Wei Sun, Jan 03 2013

Conjecture: a(n)>0 for all n>1.

This has been verified for n up to 10^9. It implies that there are infinitely many twin primes and also infinitely many cousin primes, since the interval [m!+2,m!+m] of length m-2 contains no prime for any integer m>1.

			a(92)=1 since 92=40+52 with 6*40-1, 6*40+1, 6*52+1 and 6*52+5 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..20000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A001359, A006512, A023200, A046132, A219157, A218867, A219185, A220455.

a[n_]:=a[n]=Sum[If[PrimeQ[6k-1]==True&&PrimeQ[6k+1]==True&&PrimeQ[6(n-k)+1]==True&&PrimeQ[6(n-k)+5]==True,1,0],{k,1,n-1}]
Do[Print[n," ",a[n]],{n,1,100}]

A194098 Decimal expansion of sum of reciprocals of cousin primes.

Original entry on oeis.org

1, 1, 9, 7, 0, 4, 4, 9

Offset: 1

Kausthub Gudipati, Aug 15 2011

The value is obtained by summing cousin prime pairs with values less than 2^42 (which yields 1.10633...) and a logarithmic extrapolation of Brun's constant A065421.

The estimate by [Park-Lee] is 1.197054+-7e-6. - R. J. Mathar, Feb 09 2013

Yeonyong Park, Heonsoo Lee, On the several differences between primes, J. Appl. Math. & Computing 13 (2003) vol 1-2, pp 37-51.

Prime Curios!, Sum of reciprocals of cousin primes
Marek Wolf, On the Twin and Cousin Primes, (1996) IFTUWr 909/96
Marek Wolf, Generalized Brun's constants, (1997) IFTUWr 910/97
Eric E. Weisstein, MathWorld: Cousin primes

Equals 1.1970449... = (1/7+1/11)+(1/13+1/17)+.. = Sum_{n>=2} (1/A023200(n) + 1/A046132(n)).

A220455 Number of ways to write n=x+y (x>0, y>0) with 3x-2, 3x+2 and 2xy+1 all prime.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 0, 2, 3, 2, 1, 2, 1, 1, 4, 4, 1, 2, 2, 3, 3, 2, 2, 5, 1, 4, 1, 1, 5, 4, 1, 2, 5, 5, 3, 8, 3, 6, 5, 5, 4, 4, 2, 4, 5, 3, 1, 8, 3, 4, 4, 1, 2, 8, 6, 3, 4, 5, 4, 4, 7, 1, 3, 6, 5, 7, 3, 3, 8, 2, 4, 5, 2, 6, 10, 7, 1, 5, 5, 6, 8, 6, 4, 5, 5, 7, 5, 4, 4, 11, 4, 5, 5, 5, 6, 6, 3, 1, 12, 8

Offset: 1

Zhi-Wei Sun, Dec 15 2012

nonn

Conjecture: a(n)>0 for all n>7.
This has been verified for n up to 10^8. It implies that there are infinitely many cousin primes.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Aug 06 2023
Zhi-Wei Sun also made some other similar conjectures, e.g., he conjectured that any integer n>17 can be written as x+y (x>0, y>0) with 2x-3, 2x+3 and 2xy+1 all prime, and each integer n>28 can be written as x+y (x>0, y>0) with 2x+1, 2y-1 and 2xy+1 all prime.
Both conjectures verified for n up to 10^9. - Mauro Fiorentini, Aug 06 2023

			a(25)=1 since 25=13+12 with 3*13-2, 3*13+2 and 2*13*12+1=313 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A220431, A023200, A046132, A218867, A219055, A220419, A220413, A220272, A219842, A219864, A219923.

a[n_]:=a[n]=Sum[If[PrimeQ[3k-2]==True&&PrimeQ[3k+2]==True&&PrimeQ[2k(n-k)+1]==True,1,0],{k,1,n-1}]
Do[Print[n," ",a[n]],{n,1,1000}]
apQ[{a_,b_}]:=AllTrue[{3a-2,3a+2,2a*b+1},PrimeQ]; Table[Count[Flatten[ Permutations/@ IntegerPartitions[n,{2}],1],?(apQ[#]&)],{n,100}] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, Jun 09 2018 *)

cp's OEIS Frontend

A186243 Numbers k such that 6*k-5 and 6*k-1 are both primes.

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

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A092146 Primes of the form p + 10 where p is a prime.

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A228917 Number of undirected circular permutations i_0, i_1, ..., i_n of 0, 1, ..., n such that i_0+i_1, i_1+i_2, ...,i_{n-1}+i_n, i_n+i_0 are among those k with 6*k-1 and 6*k+1 twin primes.

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A157834 Numbers n such that 3n-2 and 3n+2 are both prime.

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A176130 Lesser of a pair (p,p+4) of cousin primes whose arithmetic mean p+2 is a square number.

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A178228 Numbers k such that (k^3 - 2, k^3 + 2) is a pair of cousin primes (see A178227).

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A186243 Numbers k such that 6k-5 and 6k-1 are both primes.

A228917 Number of undirected circular permutations i_0, i_1, ..., i_n of 0, 1, ..., n such that i_0+i_1, i_1+i_2, ...,i_{n-1}+i_n, i_n+i_0 are among those k with 6k-1 and 6k+1 twin primes.