A219923 -id:A219923 - cp's OEIS frontend

A220413 Number of ways to write n=x+y (x>=0, y>=0) with x^3+2*y^3 prime.

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

Offset: 1

Zhi-Wei Sun, Dec 13 2012

nonn

Conjecture: a(n)>0 for every n=1,2,3,... Moreover, any integer n>3 not among 7, 22, 31 can be written as p+q (q>0) with p and p^3+2*q^3 both prime.
We have verified this conjecture for n up to 10^8. D. R. Heath-Brown proved in 2001 that there are infinitely many primes in the form x^3+2*y^3, where x and y are positive integers.
Zhi-Wei Sun also made the following general conjecture: For each positive odd integer m, any sufficiently large integer n can be written as x+y (x>=0, y>=0) with x^m+2*y^m prime.
When m=1, this follows from Bertrand's postulate proved by Chebyshev in 1850. For m = 5, 7, 9, 11, 13, 15, 17, 19, it suffices to require that n is greater than 46, 69, 141, 274, 243, 189, 320, 454 respectively.

			a(9)=1 since 9=7+2 with 7^3+2*2^3=359 prime.
a(22)=1 since 22=1+21 with 1^3+2*21^3=18523 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
D. R. Heath-Brown, Primes represented by x^3 + 2y^3. Acta Mathematica 186 (2001), pp. 1-84.
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A220272, A219842, A219864, A219923.

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

A220272 Number of ways to write n=x^2+y (x>0, y>0) with 2xy-1 prime.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 2, 1, 1, 3, 2, 2, 2, 1, 1, 3, 3, 1, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 1, 1, 4, 3, 1, 2, 2, 2, 5, 3, 3, 3, 4, 3, 3, 1, 3, 3, 2, 2, 4, 4, 2, 6, 2, 2, 4, 4, 2, 3, 1, 2, 5, 4, 1, 3, 3, 3, 6, 2, 3, 5, 4, 3, 3, 3, 3, 6, 3, 2, 4, 2, 3, 4, 3, 2, 5, 3, 5, 2, 1, 1, 9, 4, 3, 4, 3, 5, 3, 2, 2, 2

Offset: 1

Zhi-Wei Sun, Dec 09 2012

nonn

Conjecture: a(n)>0 for all n>2.
This has been verified for n up to 3*10^9. The author observed that for each n=3,...,3*10^9 we may even require x<(log n)^2, but Jack Brennen found that for n=4630581798 we cannot require x<(log n)^2.
The author guessed that the conjecture can be slightly refined as follows: Any integer n>2 can be written as x^2+y with 2*x*y-1 prime, where x and y are positive integers with x<=y.
Zhi-Wei Sun also made the following general conjecture: If m is a positive integer and r is 1 or -1, then any sufficiently large integer n can be written as x^2+y (x>0, y>0) with m*x*y+r prime.
For example, for (m,r)=(1,-1),(1,1),(2,1),(3,-1),(3,1),(4,-1),(4,1),(5,-1),(5,1),(6,-1),(6,1), it suffices to require that n is greater than 12782, 15372, 488, 5948, 2558, 92, 822, 21702, 6164, 777, 952 respectively.

			a(18)=1 since 18=3^2+9 with 2*3*9-1=53 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.
Zhi-Wei Sun, Re: A curious conjecture on primes, a message to Number Theory List, Dec. 12, 2012.

Cf. A219842, A219864, A219923.

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

A199920 Number of ways to write n = p+k with p, p+6, 6k-1 and 6k+1 all prime.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 1, 2, 0, 3, 1, 3, 2, 2, 2, 3, 2, 2, 1, 2, 3, 3, 3, 1, 1, 3, 2, 4, 1, 2, 2, 3, 3, 3, 2, 4, 2, 4, 3, 3, 5, 3, 3, 3, 3, 4, 5, 3, 3, 3, 3, 5, 4, 4, 3, 4, 3, 3, 2, 3, 6, 5, 4, 2, 1, 3, 5, 5, 5, 2, 2, 3, 5, 3, 5, 4, 5, 2, 3, 2, 5, 5, 6, 4, 2, 3, 3, 4, 3, 3, 5, 4, 3, 1, 1, 4, 5, 7

Offset: 1

Zhi-Wei Sun, Dec 22 2012

Conjecture: a(n)>0 for all n>11.
This implies that there are infinitely many twin primes and also infinitely many sexy primes. It has been verified for n up to 10^9. See also A199800 for a weaker version of this conjecture.
Zhi-Wei Sun also conjectured that any integer n>6 not equal to 319 can be written as p+k with p, p+6, 3k-2+(n mod 2) and 3k+2-(n mod 2) all prime.

			a(21)=1 since 21=11+10 with 11, 11+6, 6*10-1 and 6*10+1 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..50000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A001359, A006512, A023201, A199800, A219055, A219864, A219923, A002375.

a[n_]:=a[n]=Sum[If[PrimeQ[Prime[k]+6]==True&&PrimeQ[6(n-Prime[k])-1]==True&&PrimeQ[6(n-Prime[k])+1]==True,1,0],{k,1,PrimePi[n]}]
Do[Print[n," ",a[n]],{n,1,100}]
Table[Count[Table[{n-i,i},{i,n-1}],?(AllTrue[{#[[1]],#[[1]]+6,6#[[2]]-1,6#[[2]]+1},PrimeQ]&)],{n,100}] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, May 19 2015 *)

a(n)=my(s,p=2,q=3); forprime(r=5,n+5, if(r-p==6 && isprime(6*n-6*p-1) && isprime(6*n-6*p+1), s++); if(r-q==6 && isprime(6*n-6*q-1) && isprime(6*n-6*q+1), s++); p=q; q=r); s \\ Charles R Greathouse IV, Jul 31 2016

A220419 Number of ways to write n=x+y (x>0, y>0) with 2x+1, 2y-1 and x^3+2y^3 all prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Dec 14 2012

nonn

Conjecture: a(n)>0 for all n>527.
This has been verified for n up to 2*10^7. It implies the Goldbach conjecture since 2(x+y)=(2x+1)+(2y-1).
Zhi-Wei Sun also made the following similar conjectures:
(1) Each integer n>1544 can be written as x+y (x>0, y>0) with 2x-1, 2y+1 and x^3+2y^3 all prime.
(2) Any odd number n>2060 can be written as 2p+q with p, q and p^3+2((q-1)/2)^3 all prime.
(3) Every integer n>25537 can be written as p+q (q>0) with p, p-6, p+6 and p^3+2q^3 all prime.
(4) Any even number n>1194 can be written as x+y (x>0, y>0) with x^3+2y^3 and 2x^3+y^3 both prime.
(5) Each integer n>3662 can be written as x+y (x>0, y>0) with 3(xy)^3-1 and 3(xy)^3+1 both prime.
(6) Any integer n>22 can be written as x+y (x>0, y>0) with (xy)^4+1 prime. Also, any integer n>7425 can be written as x+y (x>0, y>0) with 2(xy)^4-1 and 2(xy)^4+1 both prime.
(7) Every odd integer n>1 can be written as x+y (x>0, y>0) with x^4+y^2 prime. Moreover, any odd number n>15050 can be written as p+2q with p, q and p^4+(2q)^2 all prime.
Conjectures (1) to (7) verified up to 10^6. - Mauro Fiorentini, Sep 22 2023

			a(25)=1 since 25=3+22 with 2*3+1, 2*22-1 and 3^3+2*22^3=21323 all prime.
a(26)=1 since 26=11+15 with 2*11+1, 2*15-1 and 11^3+2*15^3=8081 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..20000
D. R. Heath-Brown, Primes represented by x^3 + 2y^3. Acta Mathematica 186 (2001), pp. 1-84.
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A220413, A173587, A220272, A219842, A219864, A219923.

a[n_]:=a[n]=Sum[If[PrimeQ[2k+1]==True&&PrimeQ[2(n-k)-1]==True&&PrimeQ[k^3+2(n-k)^3]==True,1,0],{k,1,n-1}]
Do[Print[n," ",a[n]],{n,1,1000}]

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 *)

A220431 Number of ways to write n=x+y (x>0, y>0) with 3x-1, 3x+1 and xy-1 all prime.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 3, 2, 3, 2, 2, 6, 1, 3, 6, 4, 3, 3, 2, 3, 4, 3, 4, 2, 3, 3, 5, 4, 4, 7, 1, 2, 5, 1, 5, 7, 4, 2, 3, 7, 4, 7, 2, 4, 7, 4, 4, 5, 2, 5, 8, 4, 3, 3, 5, 2, 8, 5, 4, 3, 10, 7, 8, 2, 3, 5, 5, 3, 6, 3, 3, 14, 4, 3, 12, 3, 7, 7, 5, 6, 8, 7, 5, 9, 9, 4, 4, 3, 6, 10, 8

Offset: 1

Zhi-Wei Sun, Dec 14 2012

nonn

Conjecture: a(n)>0 for all n>3.
This has been verified for n up to 10^8, and it is stronger than A. Murthy's conjecture related to A109909.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Jul 26 2023
The conjecture implies the twin prime conjecture for the following reason: If x_1<...Zhi-Wei Sun also made some similar conjectures. For example, any integer n>2 not equal to 63 can be written as x+y (x>0, y>0) with 2x-1, 2x+1 and 2xy+1 all prime.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Jul 26 2023

			a(22)=1 since 22=4+18 with 3*4-1, 3*4+1 and 4*18-1 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, A220419, A220413, A173587, A220272, A219842, A219864, A219923.

a[n_]:=a[n]=Sum[If[PrimeQ[3k-1]==True&&PrimeQ[3k+1]==True&&PrimeQ[k(n-k)-1]==True,1,0],{k,1,n-1}]
Do[Print[n," ",a[n]],{n,1,1000}]

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}]

A199800 Number of ways to write n = p+q with p, 6q-1 and 6q+1 all prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Dec 21 2012

nonn

Conjecture: a(n)>0 for all n>11.
This implies the twin prime conjecture, and it has been verified for n up to 10^9.
Zhi-Wei Sun also made some similar conjectures, for example, any integer n>5 can be written as p+q with p, 2q-3 and 2q+3 all prime, and each integer n>4 can be written as p+q with p, 3q-2+(n mod 2) and 3q+2-(n mod 2) all prime.

			a(3)=1 since 3=2+1 with 2, 6*1-1 and 6*1+1 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, A220419, A220413, A173587, A220272, A219842, A219864, A219923.

a[n_]:=a[n]=Sum[If[PrimeQ[n-k]==True&&PrimeQ[6k-1]==True&&PrimeQ[6k+1]==True,1,0],{k,1,n-1}]
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

A220413 Number of ways to write n=x+y (x>=0, y>=0) with x^3+2*y^3 prime.

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A220272 Number of ways to write n=x^2+y (x>0, y>0) with 2*x*y-1 prime.

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A199920 Number of ways to write n = p+k with p, p+6, 6k-1 and 6k+1 all prime.

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A220419 Number of ways to write n=x+y (x>0, y>0) with 2x+1, 2y-1 and x^3+2y^3 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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A220431 Number of ways to write n=x+y (x>0, y>0) with 3x-1, 3x+1 and xy-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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A199800 Number of ways to write n = p+q with p, 6q-1 and 6q+1 all prime.

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A220272 Number of ways to write n=x^2+y (x>0, y>0) with 2xy-1 prime.