A220431 Number of ways to write n=x+y (x>0, y>0) with 3x-1, 3x+1 and xy-1 all prime.
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
Keywords
A220554 Number of ways to write 2n = p+q (q>0) with p, 2p+1 and (p-1)^2+q^2 all prime.
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
Keywords
Comments
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
Examples
a(16)=1 since 32=11+21 with 11, 2*11+1=23 and (11-1)^2+21^2=541 all prime.
References
- R. K. Guy, Unsolved Problems in Number Theory, 2nd Edition, Springer, New York, 2004, p. 161.
Links
- 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.
Crossrefs
Programs
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Mathematica
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}]
A229969 Number of ways to write n = x + y + z with 0 < x <= y <= z such that all the six numbers 2*x-1, 2*y-1, 2*z-1, 2*x*y-1, 2*x*z-1, 2*y*z-1 are prime.
0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 4, 4, 3, 3, 3, 3, 2, 3, 3, 3, 3, 4, 2, 7, 4, 3, 5, 3, 2, 6, 3, 4, 3, 4, 5, 3, 4, 6, 6, 3, 5, 4, 5, 6, 9, 4, 8, 4, 7, 10, 2, 6, 12, 9, 1, 7, 7, 6, 12, 10, 3, 7, 8, 8, 9, 9, 5, 3, 7, 3, 7, 3, 9, 10, 8, 6, 11, 11, 13, 15, 6, 6, 10, 15, 11, 11, 13, 8, 12, 12, 7, 10, 8, 13, 12
Offset: 1
Keywords
Comments
Conjecture: a(n) > 0 for all n > 5. Moreover, any integer n > 6 can be written as x + y + z with x among 3, 4, 6, 10, 15 such that 2*y-1, 2*z-1, 2*x*y-1, 2*x*z-1, 2*y*z-1 are prime.
We have verified this conjecture for n up to 10^6. As (2*x-1)+(2*y-1)+(2*z-1) = 2*(x+y+z)-3, it implies Goldbach's weak conjecture which has been proved.
Zhi-Wei Sun also had some similar conjectures including the following (i)-(iii):
(i) Any integer n > 6 can be written as x + y + z (x, y, z > 0) with 2*x-1, 2*y-1, 2*z-1 and 2*x*y*z-1 all prime and x among 2, 3, 4. Also, each integer n > 2 can be written as x + y + z (x, y, z > 0) with 2*x+1, 2*y+1, 2*z+1 and 2*x*y*z+1 all prime and x among 1, 2, 3.
(ii) Each integer n > 4 can be written as x + y + z with x = 3 or 6 such that 2*y+1, 2*x*y*z-1 and 2*x*y*z+1 are prime.
(iii) Every integer n > 5 can be written as x + y + z (x, y, z > 0) with x*y-1, x*z-1, y*z-1 all prime and x among 2, 6, 10. Also, any integer n > 2 not equal to 16 can be written as x + y + z (x, y, z > 0) with x*y+1, x*z+1, y*z+1 all prime and x among 1, 2, 6.
See also A229974 for a similar conjecture involving three pairs of twin primes.
Examples
a(10) = 2 since 10 = 2+2+6 = 3+3+4 with 2*2-1, 2*6-1, 2*2*2-1, 2*2*6 -1, 2*3-1, 2*4-1, 2*3*3-1, 2*3*4-1 all prime.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
- Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.
Programs
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Mathematica
a[n_]:=Sum[If[PrimeQ[2i-1]&&PrimeQ[2j-1]&&PrimeQ[2(n-i-j)-1]&&PrimeQ[2i*j-1]&&PrimeQ[2i(n-i-j)-1]&&PrimeQ[2j(n-i-j)-1],1,0],{i,1,n/3},{j,i,(n-i)/2}] Table[a[n],{n,1,100}]
A229974 Number of ways to write n = x + y + z (x, y, z > 0) with the six numbers 2*x-1, 2*x+1, 2*x*y-1, 2*x*y+1, 2*x*y*z-1, 2*x*y*z+1 all prime.
0, 0, 0, 1, 1, 4, 2, 1, 2, 4, 5, 3, 3, 8, 1, 9, 4, 6, 3, 8, 16, 8, 4, 8, 7, 3, 10, 7, 3, 14, 4, 6, 8, 13, 12, 14, 6, 8, 13, 7, 13, 15, 13, 9, 9, 10, 7, 13, 14, 7, 16, 15, 12, 8, 16, 31, 11, 6, 16, 13, 16, 15, 26, 8, 10, 17, 10, 12, 11, 17, 9, 9, 13, 18, 17, 23, 14, 10, 7, 13, 29, 13, 18, 14, 9, 19, 21, 14, 19, 14, 25, 11, 14, 18, 13, 21, 15, 26, 14, 8
Offset: 1
Keywords
Comments
Conjecture: (i) a(n) > 0 for all n > 3. Moreover, any integer n > 3 can be written as x + y + z with x among 2, 3, 6 such that {2*x*y-1, 2*x*y+1} and {2*x*y*z-1, 2*x*y*z+1} are twin prime pairs.
(ii) Each integer n > 11 can be written as x + y + z (x, y, z > 0) with x-1, x+1, x*y-1, x*y+1, x*y*z-1, x*y*z+1 all prime, moreover we may require that x is among 4, 6, 12.
(iii) Any integer n > 3 not equal to 10 can be written as x + y + z (x, y, z > 0) such that the three numbers 2*x-1, 2*x*y-1 and 2*x*y*z-1 are Sophie Germain primes, moreover we may require that x is among 2, 3, 6.
Note that part (i) or (ii) of the above conjecture implies the twin prime conjecture, while part (iii) implies that there are infinitely many Sophie Germain primes.
See also the comments of A229969 for other similar conjectures.
Examples
a(4) = 1 since 4 = 2+1+1 with 2*2-1 and 2*2+1 both prime. a(5) = 1 since 5 = 3+1+1 with 2*3-1 and 2*3+1 both prime. a(15) = 1 since 15 = 6+5+4 with 2*6-1, 2*6+1, 2*6*5-1, 2*6*5+1, 2*6*5*4-1, 2*6*5*4+1 all prime.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 1..4000
- Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.
Programs
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Mathematica
a[n_]:=Sum[If[PrimeQ[2i-1]&&PrimeQ[2i+1]&&PrimeQ[2*i*j-1]&&PrimeQ[2i*j+1]&&PrimeQ[2i*j*(n-i-j)-1]&&PrimeQ[2i*j*(n-i-j)+1],1,0],{i,1,n-2},{j,1,n-1-i}] Table[a[n],{n,1,100}]
A230037 Number of ways to write n = x + y + z (0 < x <= y <= z) such that the four pairs {6*x-1, 6*x+1}, {6*y-1, 6*y+1}, {6*z-1, 6*z+1} and {6*x*y-1, 6*x*y+1} are twin prime pairs.
0, 0, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 3, 3, 2, 2, 2, 2, 4, 3, 2, 3, 2, 5, 2, 4, 3, 4, 4, 4, 3, 3, 4, 5, 7, 4, 5, 2, 5, 4, 5, 7, 5, 5, 4, 4, 4, 6, 6, 8, 4, 5, 3, 4, 5, 6, 7, 4, 6, 2, 5, 3, 7, 8, 4, 4, 1, 4, 2, 7, 6, 3, 5, 3, 5, 4, 6, 6, 5, 4, 3, 5, 4, 5, 3, 3, 3, 6, 7, 5, 2, 4, 4, 5, 3, 6, 4, 3, 5
Offset: 1
Keywords
Comments
Conjecture: a(n) > 0 for all n > 2. Moreover, any integer n > 2 can be written as x + y + z with x = 1 or 5 such that {6*y-1, 6*y+1}, {6*z-1, 6*z+1} and {6*x*y-1, 6*x*y+1} are twin prime pairs.
We have verified this for n up to 5*10^7. It implies the twin prime conjecture.
Zhi-Wei Sun also made the following similar conjectures:
(i) Any integer n > 2 can be written as x + y + z (x, y, z > 0) with the 8 numbers 6*x-1, 6*x+1, 6*y-1, 6*y+1, 6*z-1, 6*z+1, 6*x*y-1 and 6*x*y*z-1 (or 12*x*y-1) all prime.
(ii) Each integer n > 2 can be written as x + y + z (x, y, z > 0) with the 8 numbers 6*x-1, 6*x+1, 6*y-1, 12*y-1, 6*z-1 (or 6*x*y-1), 2*(x^2+y^2)+1, 2*(x^2+z^2)+1, 2*(y^2+z^2)+1 all prime.
(iii) Any integer n > 8 can be written as x + y + z (x, y, z > 0) with x-1, x+1, y-1, y+1, x*z-1 and y*z-1 all prime.
(iv) Every integer n > 4 can be written as p + q + r (r > 0) with p, q, 2*p*q-1, 2*p*r-1 and 2*q*r-1 all prime.
(v) Any integer n > 10 can be written as x^2 + y^2 + z (x, y, z > 0) with 2*x*y-1, 2*x*z+1 and 2*y*z+1 all prime.
Examples
a(10) = 1 since 10 = 1 + 2 + 7 , and {6*1-1, 6*1+1}, {6*2-1, 6*2+1}, {6*7-1, 6*7+1} and {6*1*2-1, 6*1*2+1} are twin prime pairs.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
- Zhi-Wei Sun, Two conjectures involving six primes, a message to Number Theory List, Oct. 5, 2013.
- Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.
Programs
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Mathematica
a[n_]:=Sum[If[PrimeQ[6i-1]&&PrimeQ[6i+1]&&PrimeQ[6j-1]&&PrimeQ[6j+1]&&PrimeQ[6i*j-1] &&PrimeQ[6*i*j+1]&&PrimeQ[6(n-i-j)-1]&&PrimeQ[6(n-i-j)+1],1,0],{i,1,n/3},{j,i,(n-i)/2}] Table[a[n],{n,1,100}]
A230254 Number of ways to write n = p + q with p and (p+1)*q/2 + 1 both prime.
0, 0, 0, 1, 1, 2, 1, 2, 2, 2, 3, 2, 1, 4, 1, 2, 5, 2, 3, 2, 3, 4, 4, 3, 4, 4, 2, 2, 8, 1, 6, 6, 2, 3, 2, 3, 5, 5, 5, 1, 5, 3, 7, 5, 1, 7, 10, 1, 3, 4, 8, 5, 3, 3, 3, 5, 8, 4, 10, 2, 9, 3, 3, 4, 7, 5, 9, 5, 4, 3, 15, 4, 12, 7, 4, 5, 9, 3, 11, 4, 6, 5, 9, 5, 6, 12, 6, 5, 8, 1, 4, 8, 5, 13, 9, 2, 6, 5, 8, 4
Offset: 1
Keywords
Comments
Conjecture: a(n) > 0 for all n > 3.
We have verified this for n up to 10^8.
We also have some similar conjectures, for example, any integer n > 3 not equal to 17 or 66 can be written as p + q with p and (p+1)*q/2 - 1 both prime.
Examples
a(15) = 1 since 15 = 5 + 10 with 5 and (5+1)*10/2+1 = 31 both prime. a(30) = 1 since 30 = 2 + 28 with 2 and (2+1)*28/2+1 = 43 both prime.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
- Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.
Programs
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Mathematica
a[n_]:=Sum[If[PrimeQ[(Prime[i]+1)(n-Prime[i])/2+1],1,0],{i,1,PrimePi[n-1]}] Table[a[n],{n,1,100}]
A232186 Number of ways to write n = p + q (q > 0) with p and p^3 + n*q^2 both prime.
0, 0, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 2, 3, 2, 2, 5, 1, 1, 3, 1, 5, 4, 2, 3, 3, 1, 2, 3, 2, 4, 6, 2, 3, 5, 2, 3, 3, 3, 2, 3, 4, 2, 4, 3, 2, 2, 3, 2, 6, 2, 3, 3, 5, 4, 4, 4, 5, 9, 1, 4, 7, 3, 4, 6, 3, 5, 8, 3, 5, 6, 5, 5, 13, 2, 4, 5, 4, 4, 7, 5, 5, 13, 3, 5, 8, 6, 4, 6, 4, 3, 8, 3, 4, 9, 1, 4, 11, 3
Offset: 1
Keywords
Comments
Conjecture: a(n) > 0 for all n > 2.
Examples
a(10) = 1 since 10 = 7 + 3 with 7 and 7^3 + 10*3^2 = 433 both prime. a(11) = 1 since 11 = 5 + 6 with 5 and 5^3 + 11*6^2 = 521 both prime. a(124) = 1 since 124 = 19 + 105 with 19 and 19^3 + 124*105^2 = 1373959 both prime.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
- Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.
Programs
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Mathematica
a[n_]:=Sum[If[PrimeQ[Prime[k]^3+n*(n-Prime[k])^2],1,0],{k,1,PrimePi[n-1]}] Table[a[n],{n,1,100}]
A230261 Number of ways to write 2*n - 1 = p + q with p, p + 6 and q^4 + 1 all prime, where q is a positive integer.
0, 0, 0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 4, 3, 3, 4, 4, 3, 3, 4, 1, 5, 4, 3, 5, 5, 5, 4, 6, 4, 5, 5, 3, 3, 5, 4, 4, 2, 6, 8, 5, 4, 6, 7, 5, 5, 7, 6, 5, 7, 4, 6, 6, 3, 6, 5, 7, 6, 4, 6, 7, 6, 2, 7, 6, 2, 5, 5, 3, 7, 7, 5, 7, 9, 6, 7, 4, 6, 6, 4, 3, 9, 7, 4, 9, 9, 6, 5, 10, 8, 5, 9, 6, 7, 8, 4
Offset: 1
Keywords
Comments
Conjecture: (i) a(n) > 0 for all n > 3. Also, any odd number greater than 6 can be written as p + q (q > 0) with p, p + 6 and q^2 + 1 all prime.
(ii) Any integer n > 1 can be written as x + y (x, y > 0) with x^4 + 1 and y^2 + y + 1 both prime.
(iii) Each integer n > 2 can be expressed as x + y (x, y > 0) with 4*x^2 + 3 and 4*y^2 -3 both prime.
Either of parts (i) and (ii) implies that there are infinitely many primes of the form x^4 + 1.
Examples
a(6) = 2 since 2*6-1 = 5 + 6 = 7 + 4, and 5, 5+6 = 11, 7, 7+6 = 13, 6^4+1 = 1297 and 4^4+1 = 257 are all prime. a(25) = 1 since 2*25-1 = 47 + 2, and 47, 47+6 = 53, 2^4+1 = 17 are all prime.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
- Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.
Programs
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Mathematica
a[n_]:=Sum[If[PrimeQ[Prime[i]+6]&&PrimeQ[(2n-1-Prime[i])^4+1],1,0],{i,1,PrimePi[2n-2]}] Table[a[n],{n,1,100}] -
PARI
a(n)=my(s,p=5,q=7);forprime(r=11,2*n+4,if(r-p==6&&isprime((2*n-1-p)^4+1),s++); if(r-q==6&&isprime((2*n-1-q)^4+1),s++); p=q;q=r);s \\ Charles R Greathouse IV, Oct 14 2013
A199800 Number of ways to write n = p+q with p, 6q-1 and 6q+1 all prime.
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
Keywords
Comments
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.
Examples
a(3)=1 since 3=2+1 with 2, 6*1-1 and 6*1+1 all prime.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
- Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.
Programs
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Mathematica
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}]
A231633 Number of ways to write n = x + y (x, y > 0) with x^2 * y - 1 prime.
0, 0, 1, 2, 3, 1, 3, 2, 5, 2, 4, 2, 7, 2, 5, 3, 5, 3, 10, 4, 5, 3, 8, 3, 14, 6, 5, 4, 11, 5, 11, 3, 11, 9, 4, 5, 10, 5, 11, 9, 12, 3, 19, 7, 11, 6, 12, 9, 11, 7, 17, 7, 13, 5, 22, 3, 3, 15, 16, 5, 25, 4, 9, 11, 13, 5, 19, 6, 22, 6, 11, 6, 39, 6, 24, 7, 7, 6, 25, 8, 21, 11, 24, 7, 31, 7, 19, 11, 33, 10, 14, 8, 15, 27, 18, 9, 21, 4, 27, 9
Offset: 1
Keywords
Comments
Conjectures:
(i) a(n) > 0 for all n > 2. Also, any integer n > 4 can be written as x + y (x, y > 0) with x^2 * y + 1 prime.
(ii) Each n = 2, 3, ... can be expressed as x + y (x, y > 0) with (x*y)^2 + x*y + 1 prime.
(iii) Also, any integer n > 2 can be written as x + y (x, y > 0) with 2*(x*y)^2 - 1 (or (x*y)^2 + x*y - 1) prime.
From Mauro Fiorentini, Jul 31 2023: (Start)
Both parts of conjecture (i) verified for n up to 10^9.
Conjecture (ii) and both parts of conjecture (iii) verified for n up to 10^7. (End)
Examples
a(6) = 1 since 6 = 4 + 2 with 4^2*2 - 1 = 31 prime.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Crossrefs
Programs
-
Mathematica
a[n_]:=Sum[If[PrimeQ[x^2*(n-x)-1],1,0],{x,1,n-1}] Table[a[n],{n,1,100}]
Comments
Examples
Links
Crossrefs
Programs
Mathematica
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}]