A230140 -id:A230140 - cp's OEIS frontend

A230219 Number of ways to write 2*n + 1 = p + q + r with p <= q such that p, q, r are primes in A230217 and p + q + 9 is also prime.

0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 2, 5, 2, 2, 4, 3, 1, 4, 3, 1, 4, 1, 2, 5, 2, 3, 4, 3, 3, 8, 6, 3, 12, 6, 2, 13, 3, 3, 7, 6, 4, 5, 4, 4, 8, 7, 4, 12, 7, 3, 19, 6, 3, 16, 5, 4, 9, 5, 5, 7, 10, 4, 5, 8, 3, 14, 4, 3, 14, 2, 5, 12, 5, 2, 14, 9, 2, 10, 12, 4, 12, 7, 6, 12, 7, 9, 14, 8, 6, 12, 5, 4, 19, 8, 4, 23, 6, 3, 14

Offset: 1

Zhi-Wei Sun, Oct 11 2013

nonn

Conjecture: a(n) > 0 for all n > 6.
This implies Goldbach's weak conjecture for odd numbers and also Goldbach's conjecture for even numbers.
The conjecture also implies that there are infinitely many primes in A230217.

			a(18) = 1 since 2*18 + 1 = 7 + 13 + 17, and 7, 13, 17 are terms of A230217, and 7 + 13 + 9 = 29 is a 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. A002375, A068307, A230217, A230140, A230141.

RQ[n_]:=PrimeQ[n+6]&&PrimeQ[3n+8]
SQ[n_]:=PrimeQ[n]&&RQ[n]
a[n_]:=Sum[If[RQ[Prime[i]]&&RQ[Prime[j]]&&PrimeQ[Prime[i]+Prime[j]+9]&&SQ[2n+1-Prime[i]-Prime[j]],1,0],{i,1,PrimePi[n-1]},{j,i,PrimePi[2n-2-Prime[i]]}]
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

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}]

A230138 List of those primes p with p + 2 and 2*p - 5 both prime.

Original entry on oeis.org

5, 11, 17, 29, 59, 71, 101, 137, 149, 179, 197, 227, 281, 311, 431, 599, 617, 641, 809, 821, 857, 1151, 1277, 1319, 1451, 1481, 1487, 1607, 1667, 1697, 1997, 2027, 2081, 2111, 2129, 2339, 2657, 2711, 2789, 3167, 3329, 3371, 3461, 3557, 3767, 3917, 3929, 4049, 4217, 4259

Offset: 1

Zhi-Wei Sun, Oct 10 2013

nonn

Clearly, all terms are congruent to 5 modulo 6, and not congruent to 3 modulo 5. Primes in this sequence are sparser than twin primes and Sophie Germain primes.
This sequence is interesting because of the conjectures in the comments in A230140 and A230141.
Intersection of A001359 and A089253 (or A063909). - M. F. Hasler, Oct 10 2013

			a(1) = 5 since neither 2 + 2 nor 2*3 - 5 is prime, but 5 + 2 = 7 and 2*5 - 5 = 5 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A001359, A006512, A005384, A230140, A230141.

PQ[p_]:=PQ[p]=PrimeQ[p+2]&&PrimeQ[2p-5]
m=0
Do[If[PQ[Prime[n]],m=m+1;Print[m," ",Prime[n]]],{n,1,584}]

is_A230138(p)=isprime(p)&&isprime(p+2)&&isprime(p*2-5) \\ For large p it might be much faster to check first whether p%6==5. - M. F. Hasler, Oct 10 2013

A230224 Number of ways to write 2*n = p + q + r + s with p <= q <= r <= s such that p, q, r, s are primes in A230223.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Oct 12 2013

nonn

Conjecture: a(n) > 0 for all n > 17.

			a(21) = 1 since 2*21 = 7 + 7 + 11 + 17, and 7, 11, 17 are primes in A230223.
a(27) = 1 since 2*27 = 7 + 11 + 17 + 19, and 7, 11, 17, 19 are primes in A230223.

Zhi-Wei Sun, Table of n, a(n) for n = 1..3000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Cf. A002375, A068307, A230223, A230140, A230141, A230219.

RQ[n_]:=n>5&&PrimeQ[3n-4]&&PrimeQ[3n-10]&&PrimeQ[3n-14]
SQ[n_]:=PrimeQ[n]&&RQ[n]
a[n_]:=Sum[If[RQ[Prime[i]]&&RQ[Prime[j]]&&RQ[Prime[k]]&&SQ[2n-Prime[i]-Prime[j]-Prime[k]],1,0],
{i,1,PrimePi[n/2]},{j,i,PrimePi[(2n-Prime[i])/3]},{k,j,PrimePi[(2n-Prime[i]-Prime[j])/2]}]
Table[a[n],{n,1,100}]

A230230 Number of ways to write 2n = p + q with p, q, 3p - 10, 3*q + 10 all prime.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 4, 3, 3, 5, 1, 5, 5, 3, 4, 5, 3, 2, 6, 4, 3, 6, 3, 3, 6, 3, 5, 6, 3, 6, 5, 4, 4, 9, 5, 4, 9, 5, 3, 9, 4, 4, 6, 4, 5, 6, 5, 5, 10, 4, 8, 10, 3, 7, 12, 3, 6, 11, 5, 7, 8, 3, 4, 6, 6, 4, 7, 2, 7, 9, 2, 10, 9, 3, 9, 8, 3, 5, 14, 8, 4, 12, 5, 5, 11, 5, 6, 8, 3, 8, 7, 4, 9, 11, 3

Offset: 1

Zhi-Wei Sun, Oct 12 2013

nonn

Conjecture: a(n) > 0 for all n > 3.
This is stronger than Goldbach's conjecture for even numbers. If 2*n = p + q with p, q, 3*p - 10, 3*q + 10 all prime, then 6*n is the sum of the two primes 3*p - 10 and 3*q + 10.
Conjecture verified for 2*n up to 10^9. - Mauro Fiorentini, Jul 08 2023

			a(5) = 1 since 2*5 = 7 + 3 with 3*7 - 10 = 11 and 3*3 + 10 = 19 both prime.
a(16) = 1 since 2*16 = 13 + 19 with 3*13 - 10 = 29 and 3*19 + 10 = 67 both 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 [math.NT], 2012-2017.

Cf. A002375, A187757, A199920, A227923, A230227, A230140, A230141, A230217, A230219, A230223, A230224.

PQ[n_]:=n>3&&PrimeQ[3n-10]
SQ[n_]:=PrimeQ[n]&&PrimeQ[3n+10]
a[n_]:=Sum[If[PQ[Prime[i]]&&SQ[2n-Prime[i]],1,0],{i,1,PrimePi[2n-2]}]
Table[a[n],{n,1,100}]

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 *)

A230194 Number of ways to write n = x + y + z (x, y, z > 0) such that all the 11 integers 6x-1, 6x+1, 6x-5, 6x+5, 6y-1, 6y-5, 6y+5, 6(x+y)+5, 6z-1, 6z-5 and 6*z+5 are prime.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 3, 3, 3, 4, 3, 6, 5, 3, 3, 3, 5, 4, 4, 4, 2, 9, 10, 9, 7, 5, 12, 8, 2, 8, 6, 6, 7, 9, 4, 3, 10, 11, 2, 4, 6, 10, 9, 11, 9, 4, 10, 17, 9, 1, 4, 7, 6, 6, 6, 2, 5, 14, 13, 7, 5, 14, 6, 3, 5, 4, 12, 11, 14, 5, 2, 16, 11, 5, 9, 6, 8, 11, 23, 15, 3, 23, 18, 17, 9, 8, 20, 5, 10, 14, 3, 14, 15, 16, 9, 8, 24, 10, 7

Offset: 1

Zhi-Wei Sun, Oct 11 2013

nonn

Conjecture: a(n) > 0 for all n > 5.
Let S be the set of those primes p with p-4 and p+6 also prime. Since each element of S has the form 6*k-1 with k > 0, the conjecture implies that 6*n-3 with n > 5 can be expressed as a sum of three primes in the set S. If n = x + y + z, then 6*n = (6*(x+y)+5) + (6*z-5). So a(n) > 0 implies Goldbach's conjecture for the even number 6*n.
Let T be the set of those primes p with p+4 and p-6 also prime. Clearly each element of T has the form 6*k+1 with k > 0. We conjecture that 6*n+3 with n > 5 can be expressed as a sum of three primes in the set T.

			a(30) = 2 since 30 = 2 + 14 + 14 = 18 + 4 + 8, and 6*2-1 = 11, 6*2+1 = 13, 6*2-5 = 7, 6*2+5 = 17, 6*14-1 = 83, 6*14-5 = 79, 6*14+5 = 89, 6*(2+14)+5 = 101, 6*18-1 = 107, 6*18+1 = 109, 6*18-5 = 103, 6*18+5 = 113, 6*4-1 = 23, 6*4-5 = 19, 6*4+5 = 29, 6*(18+4)+5 = 137, 6*8-1 = 47, 6*8-5 = 43 and 6*8+5 = 53 are all prime.

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. A230140, A230141.

SQ[n_]:=PrimeQ[6n-1]&&PrimeQ[6n-5]&&PrimeQ[6n+5]
a[n_]:=Sum[If[SQ[i]&&PrimeQ[6i+1]&&SQ[j]&&PrimeQ[6(i+j)+5]&&SQ[n-i-j],1,0],{i,1,n-2},{j,1,n-1-i}]
Table[a[n],{n,1,100}]

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A230219 Number of ways to write 2*n + 1 = p + q + r with p <= q such that p, q, r are primes in A230217 and p + q + 9 is also prime.

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

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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.

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A230138 List of those primes p with p + 2 and 2*p - 5 both prime.

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A230224 Number of ways to write 2*n = p + q + r + s with p <= q <= r <= s such that p, q, r, s are primes in A230223.

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A230230 Number of ways to write 2*n = p + q with p, q, 3*p - 10, 3*q + 10 all prime.

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

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A230194 Number of ways to write n = x + y + z (x, y, z > 0) such that all the 11 integers 6*x-1, 6*x+1, 6*x-5, 6*x+5, 6*y-1, 6*y-5, 6*y+5, 6*(x+y)+5, 6*z-1, 6*z-5 and 6*z+5 are prime.

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

A230230 Number of ways to write 2n = p + q with p, q, 3p - 10, 3*q + 10 all prime.

A230194 Number of ways to write n = x + y + z (x, y, z > 0) such that all the 11 integers 6x-1, 6x+1, 6x-5, 6x+5, 6y-1, 6y-5, 6y+5, 6(x+y)+5, 6z-1, 6z-5 and 6*z+5 are prime.