A230138 -id:A230138 - cp's OEIS frontend

A230140 Number of ways to write n = x + y + z with 0 < x <= y <= z such that 6x-1, 6y-1, 6z-1 are among those primes p (terms of A230138) with p + 2 and 2p - 5 also prime.

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

Offset: 1

Zhi-Wei Sun, Oct 10 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 2, i.e., 6*n-3 with n > 2 can be expressed as a sum of three terms from A230138. Moreover, for any integer n > 12, there are three distinct positive integers x, y, z with x + y + z = n such that 6*x-1, 6*y-1, 6*z-1 are primes in A230138.
(ii) For each integer n > 12, there are three distinct positive integers x, y, z with x + y + z = n such that 6*x-1, 6*y-1, 6*z-1 are among those primes p with p + 2 and 2*p + 9 also prime.
Note that part (i) of this conjecture implies that there are infinitely many primes in A230138.
Indices k such that a(m)>a(k) for all m>k, are (2, 10, 26, 334, 439, 544, 551, 684, ...). The only sequence which has the first 5 terms within the 3 lines of data is A212067. (Certainly a coincidence.) - M. F. Hasler, Oct 10 2013

			a(10) = 1 since 10 = 2 + 3 + 5, and the three numbers 6*2-1=11, 6*3-1=17 and 6*5-1=29 are terms of A230138.

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. A068307, A230138, A230141.

SQ[n_]:=PrimeQ[6n-1]&&PrimeQ[6n+1]&&PrimeQ[12n-7]
a[n_]:=Sum[If[SQ[i]&&SQ[j]&&SQ[n-i-j],1,0],{i,1,n/3},{j,i,(n-i)/2}]
Table[a[n],{n,1,100}]

ip(x)=isprime(6*x-1) && isprime(6*x+1) && isprime(12*x-7); a(n)=sum(x=1,n\3,sum(y=x,ip(x)*(n-x)\2,ip(y) && ip(n-x-y))) \\ - M. F. Hasler, Oct 10 2013

A230141 Number of ways to write n = x + y + z with y <= z such that 6x-1, 6y-1, 6z-1 are terms of A230138 and 6(y+z)+1 is prime.

Original entry on oeis.org

0, 0, 1, 2, 2, 2, 4, 5, 3, 2, 3, 4, 4, 5, 6, 5, 3, 5, 4, 4, 2, 4, 6, 2, 3, 2, 6, 9, 8, 8, 5, 5, 4, 5, 10, 14, 10, 12, 6, 11, 7, 9, 13, 6, 11, 3, 9, 7, 8, 14, 6, 11, 4, 4, 8, 9, 15, 15, 7, 14, 3, 6, 13, 10, 19, 6, 6, 12, 5, 10, 8, 7, 16, 6, 10, 4, 7, 19, 11, 13, 3, 12, 5, 6, 13, 5, 12, 7, 8, 4, 5, 6, 10, 6, 4, 6, 4, 6, 7, 7

Offset: 1

Zhi-Wei Sun, Oct 10 2013

nonn

Conjecture: a(n) > 0 for all n > 2. Also, any integer n > 2 can be written as x + y + z (x, y, z > 0) such that 6*x-1, 6*y-1, 6*z-1 are terms of A230138 and 6*y*z-1 is prime.
This is a further refinement of the conjecture in A230140.
Note that if x + y + z = n then 6*n = (6*x-1) + (6*(y+z)+1). So a(n) > 0 implies Goldbach's conjecture for the even number 6*n.

			a(10) = 2 since 10 = 3 + 2 + 5 = 5 + 2 + 3, and 6*3-1 = 17, 6*2-1 = 11, 6*5-1 = 29 are terms of A230138, and 6*(2+5)+1 = 43 and 6*(2+3)+1 = 31 are also prime.

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, A002375, A068307, A230138, A230140.

SQ[n_]:=PrimeQ[6n-1]&&PrimeQ[6n+1]&&PrimeQ[12n-7]
a[n_]:=Sum[If[SQ[i]&&PrimeQ[6(n-i)+1]&&SQ[j]&&SQ[n-i-j],1,0],{i,1,n-2},{j,1,(n-i)/2}]
Table[a[n],{n,1,100}]

A230223 Primes p such that 3p-4, 3p-10, and 3*p-14 are all prime.

Original entry on oeis.org

7, 11, 17, 19, 31, 37, 47, 59, 79, 107, 131, 151, 157, 229, 317, 367, 409, 431, 479, 499, 521, 541, 739, 787, 1031, 1181, 1307, 1381, 1487, 1601, 1697, 1747, 1951, 2551, 2749, 2767, 2917, 3251, 3391, 3449, 3581, 3931, 4217, 4349, 4447, 4567, 4639, 4721, 4909, 4967

Offset: 1

Zhi-Wei Sun, Oct 12 2013

nonn

Conjecture: Any even number greater than 35 can be written as a sum of four terms of this sequence.

Primes in the sequence should be sparser than twin primes although this has not been proved.

			a(1) = 7 since 3*7-4 = 17, 3*7-10 = 11 and 3*7-14 = 7 are prime.

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. A023201, A046131, A230138, A230140, A230217, A230219, A230224.

RQ[n_]:=n>5&&PrimeQ[3n-4]&&PrimeQ[3n-10]&&PrimeQ[3n-14]
m=0
Do[If[RQ[Prime[n]],m=m+1;Print[m," ",Prime[n]]],{n,1,1000}]
Select[Prime[Range[700]],AllTrue[3#-{4,10,14},PrimeQ]&] (* Harvey P. Dale, Sep 29 2023 *)

is(p)=isprime(p) && isprime(3*p-4) && isprime(3*p-10) && isprime(3*p-14) \\ Charles R Greathouse IV, Oct 12 2013

A230227 Primes p with 3*p - 10 also prime.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 23, 31, 37, 41, 47, 53, 59, 61, 67, 79, 83, 89, 97, 101, 107, 109, 131, 137, 151, 157, 163, 167, 173, 191, 193, 199, 223, 229, 251, 257, 269, 277, 283, 307, 313, 317, 331, 347, 353, 367, 373, 397, 401, 409

Offset: 1

Zhi-Wei Sun, Oct 12 2013

nonn

Conjecture: For any integer n > 4 not equal to 76, we have 2*n = p + q for some terms p and q from the sequence.

This is stronger than Goldbach's conjecture for even numbers.

			a(1) = 5 since 3*5 - 10 = 5 is prime.

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. A000040, A002375, A230138, A230140, A230141, A230217, A230219, A230223, A230224, A230230.

PQ[p_]:=PQ[p]=p>3&&PrimeQ[3p-10]
m=0
Do[If[PQ[Prime[n]],m=m+1;Print[m," ",Prime[n]]],{n,1,80}]
Select[Prime[Range[100]],PrimeQ[3#-10]&] (* Harvey P. Dale, Jun 28 2015 *)

cp's OEIS Frontend

A230140 Number of ways to write n = x + y + z with 0 < x <= y <= z such that 6*x-1, 6*y-1, 6*z-1 are among those primes p (terms of A230138) with p + 2 and 2*p - 5 also prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A230141 Number of ways to write n = x + y + z with y <= z such that 6*x-1, 6*y-1, 6*z-1 are terms of A230138 and 6*(y+z)+1 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A230223 Primes p such that 3*p-4, 3*p-10, and 3*p-14 are all prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A230227 Primes p with 3*p - 10 also prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A230140 Number of ways to write n = x + y + z with 0 < x <= y <= z such that 6x-1, 6y-1, 6z-1 are among those primes p (terms of A230138) with p + 2 and 2p - 5 also prime.

A230141 Number of ways to write n = x + y + z with y <= z such that 6x-1, 6y-1, 6z-1 are terms of A230138 and 6(y+z)+1 is prime.

A230223 Primes p such that 3p-4, 3p-10, and 3*p-14 are all prime.