A230219 -id:A230219 - cp's OEIS frontend

A230507 Number of ways to write n = a + b + c with a <= b <= c, where a, b, c are among those numbers m (terms of A230506) with 2m + 1 and 2m^3 + 1 both prime.

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

Offset: 1

Zhi-Wei Sun, Oct 21 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 2.
(ii) Any integer n > 8 can be written as x + y + z (x, y, z > 0) with 2*x + 1, 2*y + 1, 2*z - 1, 2*x^4 - 1, 2*y^4 - 1, 2*z^4 - 1 all prime.
Either of the two parts of the conjecture is stronger than Goldbach's weak conjecture which was finally proved by H. Helfgott in 2013.
Part (i) implies that there are infinitely many positive integers n with 2*n + 1 and 2*n^3 + 1 both prime, and part (ii) implies that there are infinitely many positive integers n with 2*n + 1 and 2*n^4 - 1 both prime.
We have verified the conjecture for n up to 10^6.

			a(8) = 2 since 8 = 1 + 1 + 6 = 1 + 2 + 5, and 2*1 + 1 = 3, 2*1^3 + 1 = 3, 2*6 + 1 = 13, 2*6^3 + 1 = 433, 2*2 + 1 = 5, 2*2^3 + 1 = 17, 2*5 + 1 = 11, 2*5^3 + 1 = 251 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, On representations via sparse primes, a message to Number Theory List, Oct. 23, 2013.
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Cf. A000040, A068307, A230219, A230351, A230493, A230502, A230506.

pp[n_]:=PrimeQ[2n+1]&&PrimeQ[2n^3+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}]

A230516 Number of ways to write n = a + b + c with 0 < a <= b <= c such that {a^2+a-1, a^2+a+1}, {b^2+b-1, b^2+b+1}, {c^2+c-1, c^2+c+1} are twin prime pairs.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Oct 22 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 twin prime pairs of the form {x^2 + x - 1, x^2 + x + 1}.
See also A230514 for a similar conjecture.

			a(8) = 1 since 8 = 2 + 3 + 3, and {2*3 - 1, 2*3 + 1} = {5, 7} and {3*4 - 1, 3*4 + 1} = {11, 13} are twin prime pairs.
a(39) = 1 since 39 = 3 + 15 + 21, and {3*4 - 1, 3*4 + 1} = {11, 13}, {15*16 - 1, 15*16 + 1} = {239, 241}, {21*22 - 1, 21*22 + 1} = {461, 463} are twin prime pairs.

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. A001359, A006512, A088485, A227923, A230219, A230252, A230351, A230514.

pp[n_]:=PrimeQ[n(n+1)-1]&&PrimeQ[n(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}]

A230243 Number of primes p < n with 3p + 8 and (p-1)n + 1 both prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Oct 13 2013

nonn

Conjecture: a(n) > 0 for all n > 4.
This implies A. Murthy's conjecture (cf. A034693) that for any integer n > 1, there is a positive integer k < n such that k*n + 1 is prime.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Sep 21 2023

			a(8) = 1 since 8 = 3 + 5 with 3, 3*3+8 = 17, (3-1)*8+1 = 17 all prime.
a(17) = 1 since 17 = 7 + 10, and 7, 3*7+8 = 29, (7-1)*17+1 = 103 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 [math.NT], 2012-2017.

Cf. A034693, A085053, A086685, A023210, A219864, A230217, A230219, A230241.

a[n_]:=Sum[If[PrimeQ[3Prime[i]+8]&&PrimeQ[(Prime[i]-1)n+1],1,0],{i,1,PrimePi[n-1]}]
Table[a[n],{n,1,100}]

cp's OEIS Frontend

A230507 Number of ways to write n = a + b + c with a <= b <= c, where a, b, c are among those numbers m (terms of A230506) with 2m + 1 and 2m^3 + 1 both prime.

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A230516 Number of ways to write n = a + b + c with 0 < a <= b <= c such that {a^2+a-1, a^2+a+1}, {b^2+b-1, b^2+b+1}, {c^2+c-1, c^2+c+1} are twin prime pairs.

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A230243 Number of primes p < n with 3p + 8 and (p-1)n + 1 both prime.

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cp's OEIS Frontend

A230507 Number of ways to write n = a + b + c with a <= b <= c, where a, b, c are among those numbers m (terms of A230506) with 2*m + 1 and 2*m^3 + 1 both prime.

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Mathematica

A230516 Number of ways to write n = a + b + c with 0 < a <= b <= c such that {a^2+a-1, a^2+a+1}, {b^2+b-1, b^2+b+1}, {c^2+c-1, c^2+c+1} are twin prime pairs.

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Mathematica

A230243 Number of primes p < n with 3*p + 8 and (p-1)*n + 1 both prime.

Views

Author

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Comments

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Programs

Mathematica

A230507 Number of ways to write n = a + b + c with a <= b <= c, where a, b, c are among those numbers m (terms of A230506) with 2m + 1 and 2m^3 + 1 both prime.

A230243 Number of primes p < n with 3p + 8 and (p-1)n + 1 both prime.