A229969 -id:A229969 - cp's OEIS frontend

A230040 Number of ways to write n = x + y + z with y <= z such that all the five numbers 6x-1, 6y-1, 6z-1, 6x*y-1 and 6xz-1 are Sophie Germain primes.

0, 0, 1, 2, 2, 3, 3, 1, 3, 4, 5, 2, 1, 1, 3, 4, 4, 3, 4, 6, 5, 2, 2, 6, 5, 1, 2, 4, 2, 2, 3, 6, 5, 7, 6, 2, 3, 4, 4, 2, 3, 5, 1, 4, 7, 4, 6, 3, 9, 4, 2, 5, 4, 3, 9, 2, 4, 3, 6, 3, 5, 8, 8, 5, 8, 6, 2, 4, 3, 4, 1, 6, 4, 3, 8, 8, 6, 6, 9, 11, 2, 4, 2, 8, 3, 4, 6, 10, 5, 11, 7, 8, 6, 10, 4, 1, 3, 1, 3, 3

Offset: 1

Zhi-Wei Sun, Oct 06 2013

nonn

Conjecture: a(n) > 0 for all n > 2.
This implies that 6*n-3 with n > 2 can be expressed as a sum of three Sophie Germain primes (i.e., those primes p with 2*p+1 also prime).
We have verified the conjecture for n up to 10^8. Note that any Sophie Germain prime p > 3 has the form 6*k-1.

			a(4) = 2, since 4 = 1 + 1 + 2 = 2 + 1 + 1, and 6*1-1=5 and 6*2-1=11 are Sophie Germain primes.
a(26) = 1, since 26 = 15 + 2 + 9, and all the five numbers 6*15-1=89, 6*2-1=11, 6*9-1=53, 6*15*2-1=179 and 6*15*9=809 are Sophie Germain primes.

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.

Cf. A005384, A219842, A219864, A227938, A229969, A229974, A230037.

SQ[n_]:=PrimeQ[n]&&PrimeQ[2n+1]
a[n_]:=Sum[If[SQ[6i-1]&&SQ[6j-1]&&SQ[6(n-i-j)-1]&&SQ[6i*j-1]&&SQ[6*i(n-i-j)-1],1,0],{i,1,n-2},{j,1,(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 2x-1, 2x+1, 2xy-1, 2xy+1, 2xyz-1, 2xyz+1 all prime.

Original entry on oeis.org

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

Zhi-Wei Sun, Oct 05 2013

nonn

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.

			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.

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.

Cf. A001359, A006512, A005384, A229969, A219842, A219864.

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 {6x-1, 6x+1}, {6y-1, 6y+1}, {6z-1, 6z+1} and {6xy-1, 6xy+1} are twin prime pairs.

Original entry on oeis.org

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

Zhi-Wei Sun, Oct 06 2013

nonn

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.

			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.

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.

Cf. A001359, A006512, A219842, A219864, A229969, A229974, A230040.

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}]

A227920 Number of ways to write n = x + y + z with y and z distinct and greater than x such that 6x-1, 6y-1, 6xy-1 are Sophie Germain primes and {6x-1, 6x+1}, {6z-1, 6z+1}, {6xz-1, 6xz+1} are twin prime pairs.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 3, 1, 3, 1, 2, 4, 1, 3, 1, 3, 4, 1, 4, 2, 5, 4, 1, 4, 4, 3, 5, 1, 3, 2, 3, 8, 2, 6, 4, 4, 7, 2, 6, 5, 3, 8, 2, 6, 6, 3, 10, 2, 8, 4, 4, 10, 2, 9, 4, 4, 6, 1, 7, 4, 4, 8, 5, 3, 6, 4, 7, 1, 3, 5, 2, 10, 3, 7, 5, 3, 11, 3, 9, 4, 5, 6, 1, 7, 5, 5, 9, 4, 6, 4, 6, 9, 2, 5, 4, 3, 5, 2, 6

Offset: 1

Zhi-Wei Sun, Oct 08 2013

nonn

By part (i) of the conjecture in the comments in A227923, for any integer n > 5 not equal to 14 we have a(n) > 0, because there are distinct positive integers x, y, z with x = 1 such that 6*x-1, 6*y-1, 6*x*y-1 are Sophie Germain primes and {6*x-1, 6*x+1}, {6*z-1, 6*z+1}, {6*x*z-1, 6*x*z+1} are twin prime pairs.

Conjecture: Any integer n > 2 can be written as x + y + z (x, y, z > 0) such that 6*x-1, 6*y-1, 6x*y-1, 6*z-1 are Sophie Germain primes, and {6*x-1, 6*x+1}, and {6*y-1, 6*y+1} are twin prime pairs.

			a(14) = 1 since 14 = 2 + 7 + 5, and 6*2-1 = 11, 6*7-1 = 41, 6*2*7-1 = 83 are Sophie Germain primes, and {6*2-1, 6*2+1} ={11, 13}, {6*5-1, 6*5+1} = {29, 31}, {6*2*5-1, 6*2*5+1} = {59, 61} are twin prime pairs.

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.

Cf. A001359, A006512, A005384, A045536, A227923, A229969, A229974, A230037, A230040.

SQ[n_]:=PrimeQ[6n-1]&&PrimeQ[12n-1]
TQ[n_]:=PrimeQ[6n-1]&&PrimeQ[6n+1]
RQ[n_]:=TQ[n]&&PrimeQ[12n-1]
a[n_]:=Sum[If[RQ[i]&&SQ[j]&&SQ[i*j]&&TQ[n-i-j]&&TQ[i(n-i-j)]&&Abs[n-i-2j]>0,1,0],{i,1,n/3-1},{j,i+1,n-1-2i}]
Table[a[n],{n,1,100}]

cp's OEIS Frontend

A230040 Number of ways to write n = x + y + z with y <= z such that all the five numbers 6*x-1, 6*y-1, 6*z-1, 6*x*y-1 and 6*x*z-1 are Sophie Germain primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A227920 Number of ways to write n = x + y + z with y and z distinct and greater than x such that 6*x-1, 6*y-1, 6*x*y-1 are Sophie Germain primes and {6*x-1, 6*x+1}, {6*z-1, 6*z+1}, {6*x*z-1, 6*x*z+1} are twin prime pairs.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A230040 Number of ways to write n = x + y + z with y <= z such that all the five numbers 6x-1, 6y-1, 6z-1, 6x*y-1 and 6xz-1 are Sophie Germain primes.

A229974 Number of ways to write n = x + y + z (x, y, z > 0) with the six numbers 2x-1, 2x+1, 2xy-1, 2xy+1, 2xyz-1, 2xyz+1 all prime.

A230037 Number of ways to write n = x + y + z (0 < x <= y <= z) such that the four pairs {6x-1, 6x+1}, {6y-1, 6y+1}, {6z-1, 6z+1} and {6xy-1, 6xy+1} are twin prime pairs.

A227920 Number of ways to write n = x + y + z with y and z distinct and greater than x such that 6x-1, 6y-1, 6xy-1 are Sophie Germain primes and {6x-1, 6x+1}, {6z-1, 6z+1}, {6xz-1, 6xz+1} are twin prime pairs.