A230040 -id:A230040 - cp's OEIS frontend

A227923 Number of ways to write n = x + y (x, y > 0) such that 6x-1 is a Sophie Germain prime and {6y-1, 6*y+1} is a twin prime pair.

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

Offset: 1

Zhi-Wei Sun, Oct 09 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 1. Moreover, any integer n > 4 not equal to 13 can be written as x + y with x and y distinct and greater than one such that 6*x-1 is a Sophie Germain prime and {6*y-1, 6*y+1} is a twin prime pair.
(ii) Any integer n > 1 can be written as x + y (x, y > 0) such that 6*x-1 is a Sophie Germain prime, and {6*y+1, 6*y+5} is a cousin prime pair (or {6*y-1, 6*y+5} is a sexy prime pair).
Part (i) of the conjecture implies that there are infinitely many Sophie Germain primes, and also infinitely many twin prime pairs. For example, if all twin primes does not exceed an integer N > 2, and (N+1)!/6 = x + y with 6*x-1 a Sophie Germain prime and {6*y-1, 6*y+1} a twin prime pair, then (N+1)! = (6*x-1) + (6*y+1) with 1 < 6*y+1 < N+1, hence we get a contradiction since (N+1)! - k is composite for every k = 2..N.
We have verified that a(n) > 0 for all n = 2..10^8.
Conjecture verified up to 10^9. - Mauro Fiorentini, Jul 07 2023

			a(5) = 2 since 5 = 2 + 3 = 4 + 1, and 6*2-1 = 11 and 6*4-1 = 23 are Sophie Germain primes, and {6*3-1, 6*3+1} = {17, 19} and {6*1-1, 6*1+1} = {5,7} are twin prime pairs.
a(28) = 1 since 28 = 5 + 23 with 6*5-1 = 29 a Sophie Germain prime and {6*23-1, 6*23+1} = {137, 139} a twin prime pair.

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 [math.NT], 2012-2017.

Cf. A001359, A006512, A005384, A046132, A176130, A187757, A199920, A227920, A230037, A230040.

SQ[n_]:=PrimeQ[6n-1]&&PrimeQ[12n-1]
TQ[n_]:=PrimeQ[6n-1]&&PrimeQ[6n+1]
a[n_]:=Sum[If[SQ[i]&&TQ[n-i],1,0],{i,1,n-1}]
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}]

A230514 Number of ways to write n = a + b + c (0 < a <= b <= c) such that all the three numbers a(a+1)-1, b(b+1)-1, c*(c+1)-1 are Sophie Germain primes.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Oct 21 2013

nonn

Conjecture: a(n) > 0 for all n > 5.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Sep 22 2023
This implies that there are infinitely many Sophie Germain primes of the form x^2 + x - 1.
See also A230516 for a similar conjecture.

			a(10) = 2 since 10 = 2 + 2 + 6 = 2 + 3 + 5, and 2*3 - 1 = 5, 6*7 - 1 = 41, 3*4 - 1 = 11, 5*6 - 1 = 29 are all Sophie Germain primes.
a(39) = 1 since 39 = 9 + 15 + 15, and both 9*10 - 1 = 89 and 15*16 - 1 = 239 are Sophie Germain primes.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A000040, A005384, A209253, A227923, A230040, A230515, A230516.

pp[n_]:=PrimeQ[n(n+1)-1]&&PrimeQ[2n(n+1)-1]
a[n_]:=Sum[If[pp[i]&&pp[j]&&pp[n-i-j],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}]

A227938 List of those numbers which can be written as x + y + z (x, y, z > 0) such that all the six numbers 6x-1, 6y-1, 6z-1, 6x*y-1, 6xz-1 and 6yz-1 are Sophie Germain primes.

Original entry on oeis.org

3, 4, 5, 6, 7, 9, 10, 11, 15, 16, 17, 18, 19, 20, 21, 24, 25, 28, 31, 32, 33, 34, 35, 41, 42, 44, 45, 46, 47, 49, 51, 53, 55, 58, 61, 62, 63, 64, 65, 66, 72, 74, 75, 76, 77, 78, 79, 80, 84, 86, 87, 88, 89, 90, 91, 92, 93, 94, 101, 102

Offset: 1

Zhi-Wei Sun, Oct 07 2013

nonn

This sequence is motivated by the author's conjecture in the comments in A230040.

Conjecture: a(n) < 2*n for all n > 2.

			a(1) = 3 since 3 = 1 + 1 + 1, and 6*1-1=5 is a Sophie Germain prime.
a(7) = 10 since 10 = 1 + 2 + 7, and 6*1-1=5, 6*2-1=11, 6*7-1=41, 6*1*2-1=11, 6*1*7-1=41, 6*2*7-1=83 are Sophie Germain primes.

Zhi-Wei Sun, Table of n, a(n) for n = 1..2000

Cf. A005384, A230040.

m=0
SQ[n_]:=SQ[n]=PrimeQ[n]&&PrimeQ[2n+1]
Do[Do[If[SQ[6i-1]&&SQ[6j-1]&&SQ[6(n-i-j)-1]&&SQ[6i*j-1]&&SQ[6*i(n-i-j)-1]&&SQ[6*j(n-i-j)-1],
m=m+1;Print[m," ",n];Goto[aa]],{i,1,n/3},{j,i,(n-i)/2}];
Label[aa];Continue,{n,1,102}]
sgpQ[{x_,y_,z_}]:=AllTrue[{6x-1,6y-1,6z-1,6x y-1,6x z-1,6y z-1,2(6x-1)+1,2(6y-1)+1,2(6z-1)+ 1,2(6x y-1)+1,2(6x z-1)+1,2(6y z-1)+1},PrimeQ]; Select[Total/@Select[Tuples[Range[100],3],sgpQ]//Union,#<110&] (* Harvey P. Dale, Jul 23 2024 *)

cp's OEIS Frontend

A227923 Number of ways to write n = x + y (x, y > 0) such that 6*x-1 is a Sophie Germain prime and {6*y-1, 6*y+1} is a twin prime pair.

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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.

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A230514 Number of ways to write n = a + b + c (0 < a <= b <= c) such that all the three numbers a*(a+1)-1, b*(b+1)-1, c*(c+1)-1 are Sophie Germain primes.

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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.

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A227938 List of those numbers which can be written as x + y + z (x, y, z > 0) such that all the six numbers 6*x-1, 6*y-1, 6*z-1, 6*x*y-1, 6*x*z-1 and 6*y*z-1 are Sophie Germain primes.

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A227923 Number of ways to write n = x + y (x, y > 0) such that 6x-1 is a Sophie Germain prime and {6y-1, 6*y+1} is a twin prime pair.

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.

A230514 Number of ways to write n = a + b + c (0 < a <= b <= c) such that all the three numbers a(a+1)-1, b(b+1)-1, c*(c+1)-1 are Sophie Germain primes.

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.

A227938 List of those numbers which can be written as x + y + z (x, y, z > 0) such that all the six numbers 6x-1, 6y-1, 6z-1, 6x*y-1, 6xz-1 and 6yz-1 are Sophie Germain primes.