A218867 -id:A218867 - cp's OEIS frontend

A219185 Number of prime pairs {p,q} (p>q) with 3(p-q)-1 and 3(p-q)+1 both prime such that p+(1+(n mod 2))q=n.

0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 2, 1, 1, 1, 2, 1, 0, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 2, 0, 1, 2, 2, 0, 2, 2, 0, 2, 1, 0, 3, 1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 4, 1, 1, 1, 0, 1, 1, 2, 1, 1, 3, 1, 5, 2, 1, 2, 1, 0, 2, 0, 2, 3, 4, 2, 3, 3, 2, 2, 1, 3, 2, 1, 1, 2, 0, 0, 2, 1, 3, 2, 3

Offset: 1

Zhi-Wei Sun, Nov 13 2012

nonn

Conjecture: a(n)>0 for all odd n>4676 and even n>30986.

This conjecture has been verified for n up to 5*10^7. It implies Goldbach's conjecture, Lemoine's conjecture and the twin prime conjecture.

			a(11)=1 since 11=5+2*3, and both 3(5-3)-1=5 and 3(5-3)+1=7 are prime.
a(16)=2 since 16=11+5=13+3, and 3(11-5)-1, 3(11-5)+1, 3(13-3)-1, 3(13-3)+1 are 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.

Cf. A001359, A006512, A002375, A046927, A219157, A219055, A218867, A218754, A218825, A219052.

a[n_]:=a[n]=Sum[If[PrimeQ[n-(1+Mod[n,2])Prime[k]]==True&&PrimeQ[3(n-(2+Mod[n,2])Prime[k])-1]==True&&PrimeQ[3(n-(2+Mod[n,2])Prime[k])+1]==True,1,0],
{k,1,PrimePi[(n-1)/(2+Mod[n,2])]}]
Do[Print[n," ",a[n]],{n,1,100000}]

a(n)=if(n%2, aOdd(n), aEven(n))
aOdd(n)=my(s); forprime(q=2,(n-1)\3, my(p=n-2*q); if(isprime(n-2*q) && isprime(3*n-9*q-1) && isprime(3*n-9*q+1), s++)); s
aEven(n)=my(s); forprime(q=2,n/2, if(isprime(n-q) && isprime(3*n-6*q-1) && isprime(3*n-6*q+1), s++)); s
\\ Charles R Greathouse IV, Jul 31 2016

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}]

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 *)

A220554 Number of ways to write 2n = p+q (q>0) with p, 2p+1 and (p-1)^2+q^2 all prime.

Original entry on oeis.org

0, 2, 3, 2, 2, 2, 2, 3, 3, 3, 1, 1, 2, 3, 3, 1, 2, 3, 4, 3, 4, 2, 2, 2, 3, 1, 3, 3, 5, 3, 1, 2, 2, 2, 5, 2, 1, 2, 2, 5, 1, 2, 4, 3, 4, 4, 3, 5, 4, 4, 1, 2, 2, 2, 4, 4, 4, 4, 6, 6, 4, 2, 6, 4, 4, 4, 2, 2, 5, 6, 3, 2, 3, 5, 5, 4, 3, 2, 4, 4, 2, 4, 4, 4, 4, 3, 4, 3, 5, 6, 3, 4, 5, 5, 3, 1, 2, 5, 3, 4

Offset: 1

Zhi-Wei Sun, Dec 15 2012

nonn

Conjecture: a(n)>0 for all n>1.
This has been verified for n up to 2*10^8. It implies that there are infinitely many Sophie Germain primes.
Note that Ming-Zhi Zhang asked (before 1990) whether any odd integer greater than 1 can be written as x+y (x,y>0) with x^2+y^2 prime, see A036468.
Zhi-Wei Sun also made the following related conjectures:
(1) Any integer n>2 can be written as x+y (x,y>=0) with 3x-1, 3x+1 and x^2+y^2-3(n-1 mod 2) all prime.
(2) Each integer n>3 not among 20, 40, 270 can be written as x+y (x,y>0) with 3x-2, 3x+2 and x^2+y^2-3(n-1 mod 2) all prime.
(3) Any integer n>4 can be written as x+y (x,y>0) with 2x-3, 2x+3 and x^2+y^2-3(n-1 mod 2) all prime. Also, every n=10,11,... can be written as x+y (x,y>=0) with x-3, x+3 and x^2+y^2-3(n-1 mod 2) all prime.
(4) Any integer n>97 can be written as p+q (q>0) with p, 2p+1, n^2+pq all prime. Also, each integer n>10 can be written as p+q (q>0) with p, p+6, n^2+pq all prime.
(5) Every integer n>3 different from 8 and 18 can be written as x+y (x>0, y>0) with 3x-2, 3x+2 and n^2-xy all prime.
All conjectures verified for n up to 10^9. - Mauro Fiorentini, Sep 21 2023

			a(16)=1 since 32=11+21 with 11, 2*11+1=23 and (11-1)^2+21^2=541 all prime.

R. K. Guy, Unsolved Problems in Number Theory, 2nd Edition, Springer, New York, 2004, p. 161.

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. A005384, A005385, A036468, A220455, A220431, A218867, A219055, A220419, A220413, A220272, A219842, A219864, A219923.

a[n_]:=a[n]=Sum[If[PrimeQ[p]==True&&PrimeQ[2p+1]==True&&PrimeQ[(p-1)^2+(2n-p)^2]==True,1,0],{p,1,2n-1}]
Do[Print[n," ",a[n]],{n,1,1000}]

A187759 Number of ways to write n=x+y (0

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 0, 2, 1, 3, 2, 1, 2, 1, 2, 2, 2, 2, 3, 1, 3, 1, 2, 3, 2, 6, 1, 3, 1, 2, 4, 3, 4, 4, 1, 3, 1, 3, 5, 2, 6, 1, 3, 2, 2, 5, 2, 5, 2, 3, 1, 2, 3, 5, 2, 4, 0, 0, 3, 1, 6, 2, 3, 3, 1, 5, 1, 5, 3, 3, 3, 1, 4, 2, 3, 3, 0, 3, 3, 3, 4, 1, 3, 1, 2, 3, 2, 4, 2, 2, 3

Offset: 1

Zhi-Wei Sun, Jan 03 2013

nonn

Conjecture: If n>200 is not among 211, 226, 541, 701, then a(n)>0.
This essentially follows from the conjecture related to A219157, since n=x+y for some positive integers x and y with 6x-1,6x+1,6y-1,6y+1 all prime if and only if 6n=p+q for some twin prime pairs {p,p-2} and {q,q+2}.
Similarly, the conjecture related to A218867 implies that any integer n>491 can be written as x+y (0A219055 implies that any integer n>1600 not among 2729 and 4006 can be written as x+y (0

			a(9)=1 since 9=2+7 with 6*2-1, 6*2+1, 6*7-1 and 6*7+1 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, A219157, A219185, A199920, A187757, A187758, A218867, A219055.

a[n_]:=a[n]=Sum[If[PrimeQ[6k-1]==True&&PrimeQ[6k+1]==True&&PrimeQ[6(n-k)-1]==True&&PrimeQ[6(n-k)+1]==True,1,0],{k,1,(n-1)/2}]
Do[Print[n," ",a[n]],{n,1,100}]

a(n)=sum(x=1,(n-1)\2,isprime(6*x-1)&&isprime(6*x+1)&&isprime(6*n-6*x-1)&&isprime(6*n-6*x+1)) \\ Charles R Greathouse IV, Feb 28 2013

A187758 Number of ways to write n=x+y (x,y>0) with 2x-3, 2x+3, 6y+1 and 6y+5 all prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 03 2013

nonn

Conjecture: a(n)>0 for all n>4.

This has been verified for n up to 10^8. It implies that there are infinitely many cousin primes and also infinitely many sexy primes.

			a(5)=1 since 5=4+1 with 2*4-3, 2*4+3, 6*1+1 and 6*1+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. A023200, A046132, A023201, A002375, A187757, A199920, A219055, A218867, A220455.

a[n_]:=a[n]=Sum[If[PrimeQ[2k-3]==True&&PrimeQ[2k+3]==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}]

A219558 Number of odd prime pairs {p,q} (p>q) such that p+(1+(n mod 2))q=n and ((p-1-(n mod 2))/q)=((q+1)/p)=1 where (-) denotes the Legendre symbol.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 3, 0, 2, 0, 1, 1, 1, 1, 2, 2, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 3, 0, 1, 1, 1, 0, 1, 1, 2, 1, 1, 2, 0, 0, 2, 1, 2, 1, 1, 0, 1, 1, 2, 2, 3, 0, 0, 0, 0

Offset: 1

Zhi-Wei Sun, Nov 23 2012

nonn

For any integer m, define s(m) as the smallest positive integer s such that for each n=s,s+1,... there are primes p>q>2 with p+(1+(n mod 2))q=n and ((p-(1+(n mod 2))m)/q)=((q+m)/p)=1. If such a positive integer s does not exist, then we set s(m)=0.
Zhi-Wei Sun has the following general conjecture: s(m) is always positive. In particular, s(0)=1239,
  s(1)=1470, s(-1)=2192, s(2)=1034, s(-2)=1292,
  s(3)=1698, s(-3)=1788, s(4)=848, s(-4)=1458,
  s(5)=1490, s(-5)=2558, s(6)=1115, s(-6)=1572,
  s(7)=1550, s(-7)=932,  s(8)=825, s(-8)=2132,
  s(9)=1154, s(-9)=1968, s(10)=1880, s(-10)=1305,
  s(11)=1052, s(-11)=1230, s(12)=2340, s(-12)=1428,
  s(13)=2492, s(-13)=2673, s(14)=1412, s(-14)=1638,
  s(15)=1185, s(-15)=1230, s(16)=978, s(-16)=1605,
  s(17)=1154, s(-17)=1692, s(18)=1757, s(-18)=2292,
  s(19)=1230, s(-19)=2187, s(20)=2048, s(-20)=1372,
  s(21)=1934, s(-21)=1890, s(22)=1440, s(-22)=1034,
  s(23)=1964, s(-23)=1322, s(24)=1428, s(-24)=2042,
  s(25)=1734, s(-25)=1214, s(26)=1260, s(-26)=1230,
  s(27)=1680, s(-27)=1154, s(28)=1652, s(-28)=1808,
  s(29)=1112, s(-29)=1670, s(30)=1820, s(-30)=1284.
Note that s(1)=1470 means that a(n)>0 for all n=1470,1471,... That s(0)=1239 is related to a conjecture of Olivier Gérard and Zhi-Wei Sun.
If we replace ((p-1-(n mod 2))/q)=((q+1)/p)=1 in the definition of a(n) by ((p-1)/q)=((q+1)/p)=1, then the new a(n) seems positive for any n>1181.

			a(14)=1 since 14=11+3 with ((11-1)/3)=((3+1)/11)=1.
a(31)=1 since 31=17+2*7 with ((17-2)/7)=((7+1)/17)=1.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Olivier Gérard and Zhi-Wei Sun, Refining Goldbach's conjecture by using quadratic residues, a message to Number Theory List, Nov. 19, 2012.
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A002375, A046927, A219055, A219157, A218867, A219185, A218754, A218825, A219052.

a[n_]:=a[n]=Sum[If[PrimeQ[n-(1+Mod[n,2])Prime[k]]==True&&JacobiSymbol[n-(1+Mod[n,2])(Prime[k]+1),Prime[k]]==1&&JacobiSymbol[Prime[k]+1,n-(1+Mod[n,2])Prime[k]]==1,1,0],{k,2,PrimePi[(n-1)/(2+Mod[n,2])]}]
Do[Print[n," ",a[n]],{n,1,10000}]

A220091 Number of ways to write n=p+q+(n mod 2)q with p>q and p, q, 6q-1, 6q+1 all prime.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 2, 1, 2, 2, 2, 3, 2, 2, 2, 3, 1, 2, 1, 1, 1, 3, 2, 2, 3, 2, 1, 1, 2, 2, 1, 2, 2, 2, 3, 3, 3, 1, 2, 3, 2, 3, 3, 1, 1, 4, 2, 1, 3, 1, 1, 3, 4, 3, 3, 2, 1, 1, 3, 3, 1, 2, 2, 4, 4, 5, 3, 1, 1, 3, 2, 3, 3, 2, 2, 4, 2, 3, 3, 0, 1, 5, 2, 2, 3, 1, 0, 2, 3

Offset: 1

Zhi-Wei Sun, Dec 04 2012

nonn

Conjecture: a(n)>0 for all even n>=8070 and odd n>=18680.
This conjecture unifies the twin prime conjecture, Goldbach's conjecture and Lemoine's conjecture. It has been verified for n up to 10^7.
Zhi-Wei Sun also made the following conjecture: Any integer n>=6782 can be written as p+q+(n mod 2)q with p>q and p, q, q-6, q+6 all prime, and any integer n>=4410 can be written as p+q+(n mod 2)q with p>q and p, q, 2q-3, 2q+3 all prime, and any integer n>=16140 can be written as p+q+(n mod 2)q with p>q and p, q, 3q-2, 3q+2 all prime.

			a(31)=1 since 31=17+2*7 with 6*7-1 and 6*7+1 twin primes.
a(32)=1 since 32=29+3 with 6*3-1 and 6*3+1 twin primes.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A001359, A006512, A002375, A046927, A219185, A219157, A218867, A219055, A219966.

a[n_]:=a[n]=Sum[If[PrimeQ[6Prime[k]-1]==True&&PrimeQ[6Prime[k]+1]==True&&PrimeQ[n-(1+Mod[n,2])Prime[k]]==True,1,0],{k,1,PrimePi[(n-1)/(2+Mod[n,2])]}]
Do[Print[n," ",a[n]],{n,1,100}]

A220572 Number of ways to write 2n-1=x+y (x,y>=0) with x^18+3*y^18 prime.

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 4, 5, 4, 1, 2, 4, 1, 4, 1, 2, 1, 2, 1, 6, 1, 4, 2, 4, 3, 6, 3, 2, 4, 2, 5, 6, 4, 5, 4, 5, 5, 8, 7, 4, 7, 7, 6, 7, 4, 6, 7, 5, 6, 3, 11, 7, 1, 5, 3, 5, 6, 6, 10, 4, 13, 12, 9, 4, 9, 10, 5, 8, 3, 6, 7, 5, 4, 8, 13, 6, 3, 5, 5, 11, 6, 13, 4, 9, 10, 8, 12, 11, 8, 7, 10, 8, 7, 8, 8

Offset: 1

Zhi-Wei Sun, Dec 16 2012

nonn

Conjecture: a(n)>0 for every n=1,2,3,.... Moreover, any odd integer greater than 2092 can be written as x+y (x,y>0) with x-3, x+3 and x^18+3*y^18 all prime.
This has been verified for n up to 2*10^6.
Zhi-Wei Sun also made the following general conjecture: For each positive integer m, any sufficiently large odd integer n can be written as x+y (x,y>0) with x-3, x+3 and x^m+3*y^m all prime (and hence there are infinitely many primes in the form x^m+3*y^m). In particular, for m = 1, 2, 3, 4, 5, 6, 18 any odd integer greater than one can be written as x+y (x,y>0) with x^m+3*y^m prime, and for m =1, 2, 3 any odd integer n>15 can be written as x+y (x,y>0) with x-3, x+3 and x^m+3*y^m all prime.
Our computation suggests that for each m=7,...,20 any odd integer greater than 32, 10, 24, 30, 48, 36, 72, 146, 48, 48, 152, 2, 238, 84 respectively can be written as x+y (x,y>0) with x^m+3*y^m prime.

			a(3)=1 since 2*3-1=5=1+4 with 1^18+3*4^18=206158430209 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A220554, A036468, A220455, A220431, A218867, A219055, A220419, A220413, A220272, A219842, A219864, A219923.

a[n_]:=a[n]=Sum[If[PrimeQ[k^18+3*(2n-1-k)^18]==True,1,0],{k,0,2n-1}]
Do[Print[n," ",a[n]],{n,1,100}]

Showing 1-9 of 9 results.

cp's OEIS Frontend

A219185 Number of prime pairs {p,q} (p>q) with 3(p-q)-1 and 3(p-q)+1 both prime such that p+(1+(n mod 2))q=n.

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

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

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A220554 Number of ways to write 2n = p+q (q>0) with p, 2p+1 and (p-1)^2+q^2 all prime.

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A187759 Number of ways to write n=x+y (0

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A187758 Number of ways to write n=x+y (x,y>0) with 2x-3, 2x+3, 6y+1 and 6y+5 all prime.

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A219558 Number of odd prime pairs {p,q} (p>q) such that p+(1+(n mod 2))q=n and ((p-1-(n mod 2))/q)=((q+1)/p)=1 where (-) denotes the Legendre symbol.

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A220091 Number of ways to write n=p+q+(n mod 2)q with p>q and p, q, 6q-1, 6q+1 all prime.

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