A230219 -id:A230219 - cp's OEIS frontend

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

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

A230217 List of those primes p with p + 6 and 3*p + 8 also prime.

Original entry on oeis.org

5, 7, 11, 13, 17, 31, 41, 47, 53, 61, 73, 83, 101, 103, 131, 151, 157, 167, 193, 223, 251, 263, 271, 277, 307, 311, 347, 367, 433, 563, 571, 593, 601, 613, 641, 647, 677, 733, 823, 857, 977, 1097, 1117, 1217, 1223, 1231, 1291, 1301, 1361, 1427

Offset: 1

Zhi-Wei Sun, Oct 11 2013

nonn

Clearly, no term is congruent to 4 modulo 5.

This sequence is interesting because of the conjecture in the comments in A230219.

			a(1) = 5 since neither 2 + 6 nor 3 + 6 is prime, but 5 + 6 = 11 and 3*5 + 8 = 23 are both prime.

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

Cf. A023201, A230219.

PQ[p_]:=PrimeQ[p+6]&&PrimeQ[3p+8]
m=0
Do[If[PQ[Prime[n]],m=m+1;Print[m," ",Prime[n]]],{n,1,225}]
Select[Prime[Range[300]],AllTrue[{#+6,3#+8},PrimeQ]&] (* Harvey P. Dale, Sep 01 2023 *)

A230502 Number of ways to write n = (2-(n mod 2))*p + q + r with p <= q <= r such that p, q, r, p^2 - 2, q^2 - 2, r^2 - 2 are all prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Oct 21 2013

nonn

Conjecture: a(n) > 0 for all n > 6.
This is stronger than Goldbach's weak conjecture which was finally proved by H. Helfgott in 2013. It also implies that there are infinitely many primes p with p^2 - 2 also prime.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Sep 22 2023

			a(10) = 1 since 10 = 2*2 + 3 + 3 with 2, 3, 2^2 - 2 = 2, 3^2 - 2 = 7 all prime.
a(19) = 2 since 19 = 3 + 3 + 13 = 5 + 7 + 7 with 3, 13, 5, 7, 3^2 - 2 = 7, 13^2 - 2 = 167, 5^2 - 2 = 23, 7^2 - 2 = 47 all 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. A000040, A062326, A068307, A230219, A230351, A230493, A230494.

pp[n_]:=PrimeQ[n^2-2]
pq[n_]:=PrimeQ[n]&&pp[n]
a[n_]:=Sum[If[pp[Prime[i]]&&pp[Prime[j]]&&pq[n-(2-Mod[n,2])Prime[i]-Prime[j]],1,0],{i,1,PrimePi[n/(4-Mod[n,2])]},{j,i,PrimePi[(n-(2-Mod[n,2])Prime[i])/2]}]
Table[a[n],{n,1,100}]

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}]

A230493 Number of ways to write n = (2-(n mod 2))p + q + r with p <= q <= r such that p, q, r, 2p^2 - 1, 2q^2 - 1, 2r^2 - 1 are all prime.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 3, 3, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 3, 3, 3, 2, 2, 3, 3, 2, 2, 2, 1, 1, 2, 2, 1, 3, 3, 1, 3, 2, 4, 1, 2, 2, 4, 3, 3, 2, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 2, 2, 2, 3, 3, 2, 4, 3, 2, 3, 5, 1, 4, 3, 3, 2, 4, 4, 3, 4, 5, 2, 4, 5, 4, 3, 2, 4, 4, 3, 2

Offset: 1

Zhi-Wei Sun, Oct 20 2013

nonn

Conjecture:  a(n) > 0 for all n > 6.
This is stronger than Goldbach's weak conjecture which was finally proved by H. Helfgott in 2013. It also implies that there are infinitely many primes p with 2*p^2 - 1 also prime.
We have verified the conjecture for n up to 10^6.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Sep 22 2023
See also A230351, A230494 and A230502 for similar conjectures.

			a(14) = 1 since 14 = 2*2 + 3 + 7 with 2, 3, 7, 2*2^2 - 1 = 7, 2*3^2 - 1 = 17, 2*7^2 - 1 = 97 all prime.
a(19) = 1 since 19 = 3 + 3 + 13, and 3, 13, 2*3^2 - 1 = 17 and 2*13^2 - 1 = 337 are all prime.
a(53) = 1 since 53 = 3 + 7 + 43, and all the six numbers 3, 7, 43, 2*3^2 - 1 = 17, 2*7^2 - 1 = 97, 2*43^2 - 1 = 3697 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 [math.NT], 2012-2017.

Cf. A000040, A068307, A106483, A230219, A230351, A230494, A230502.

pp[n_]:=PrimeQ[2n^2-1]
pq[n_]:=PrimeQ[n]&&pp[n]
a[n_]:=Sum[If[pp[Prime[i]]&&pp[Prime[j]]&&pq[n-(2-Mod[n,2])Prime[i]-Prime[j]],1,0],{i,1,PrimePi[n/(4-Mod[n,2])]},{j,i,PrimePi[(n-(2-Mod[n,2])Prime[i])/2]}]
Table[a[n],{n,1,100}]

A236832 Number of ways to write 2*n - 1 = p + q + r (p <= q <= r) with p, q and r terms of A234695.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 31 2014

nonn

Conjecture: a(n) > 0 for all n > 3.

This is stronger than Goldbach's weak conjecture which was finally proved by H. A. Helfgott in 2013.

			a(4) = 1 since 2*4 - 1 = 2 + 2 + 3 with 2 and 3 terms of A234695.
a(5) = 2 since 2*5 - 1 = 2 + 2 + 5 = 3 + 3 + 3 with 2, 3, 5 terms of A234695.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
H. A. Helfgott, Minor arcs for Goldbach's problem, arXiv:1205.5252 [math.NT], 2012-2013.
H. A. Helfgott, Major arcs for Goldbach's theorem, arXiv:1305.2897 [math.NT], 2013-2014.
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2016.

Cf. A000040, A068307, A230219, A234695, A235189.

p[n_]:=PrimeQ[Prime[n]-n+1]
q[n_]:=PrimeQ[n]&&p[n]
a[n_]:=Sum[If[p[Prime[i]]&&p[Prime[j]]&&q[2n-1-Prime[i]-Prime[j]],1,0],{i,1,PrimePi[(2n-1)/3]},{j,i,PrimePi[(2n-1-Prime[i])/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 *)

A237291 Number of ways to write 2*n - 1 = p + q + r (p <= q <= r) with p, q, r, pi(p), pi(q), pi(r) all prime, where pi(x) denotes the number of primes not exceeding x (A000720).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 0, 1, 2, 0, 2, 3, 1, 0, 2, 2, 1, 3, 2, 1, 1, 1, 1, 2, 3, 2, 2, 0, 3, 4, 2, 2, 3, 2, 1, 3, 4, 1, 5, 2, 1, 2, 3, 4, 3, 1, 1, 3, 2, 2, 4, 3, 2, 3, 3, 1, 5, 5, 1, 3, 4, 2, 3, 4, 4, 2, 4, 2, 3, 4, 2, 2

Offset: 1

Zhi-Wei Sun, Feb 06 2014

nonn

Conjecture: a(n) > 0 for all n > 36.

This is stronger than Goldbach's weak conjecture finally proved by H. A. Helfgott in 2013.

			a(16) = 1 since 2*16 - 1 = 3 + 11 + 17 with 3, 11, 17, pi(3) = 2, pi(11) = 5 and pi(17) = 7 all prime.
a(179) = 1 since 2*179 - 1 = 83 + 83 + 191 with 83, 191, pi(83) = 23 and pi(191) = 43 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
H. A. Helfgott, Minor arcs for Goldbach's problem, arXiv:1205.5252, 2012.
H. A. Helfgott, Major arcs for Goldbach's theorem, arXiv:1305.2897, 2013.
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A000720, A006450, A068307, A230219, A236832, A237284.

p[n_]:=PrimeQ[n]&&PrimeQ[PrimePi[n]]
a[n_]:=Sum[If[p[2n-1-Prime[Prime[i]]-Prime[Prime[j]]],1,0],{i,1,PrimePi[PrimePi[(2n-1)/3]]},{j,i,PrimePi[PrimePi[(2n-1-Prime[Prime[i]])/2]]}]
Table[a[n],{n,1,80}]

A230451 Number of ways to write n = x + y + z (x, y, z > 0) such that 2x + 1, 2y + 3, 2z + 5 are all prime and xy*z is a triangular number.

Original entry on oeis.org

0, 0, 1, 0, 2, 3, 0, 4, 3, 1, 7, 3, 2, 3, 7, 4, 5, 6, 3, 4, 8, 5, 8, 3, 6, 8, 9, 9, 5, 12, 2, 11, 4, 4, 4, 13, 5, 9, 13, 8, 14, 8, 3, 15, 7, 8, 10, 6, 5, 17, 15, 4, 6, 9, 8, 10, 15, 9, 7, 15, 11, 5, 6, 11, 14, 14, 7, 11, 3, 12, 23, 16, 5, 20, 14, 4, 9, 14, 5, 19, 19, 4, 3, 12, 7, 16, 5, 12, 6, 11, 12, 12, 23, 14, 23, 12, 5, 17, 14, 5

Offset: 1

Zhi-Wei Sun, Oct 19 2013

nonn

Conjecture: (i) a(n) > 0 except for n = 1, 2, 4, 7.
(ii) Any integer n > 7 can be written as x + y + z (x, y, z > 0) such that 2*x + 1, 2*y + 1, 2*x*y + 1 are primes and x*y*z is a triangular number.
(iii) Each integer n > 4 not equal to 7 or 14 can be expressed as p + q + r (p, q, r > 0) with p and 2*q + 1 both primes, and p*q*r a triangular number.
(iv) Any integer n > 6 not among 16, 20, 60 can be written as x + y + z (x, y, z > 0) such that x*y + x*z + y*z is a triangular number.
Part (i) is stronger than Goldbach's weak conjecture which was finally proved by H. Helfgott in 2013.
See also A227877 and A230596 for some related conjectures.

			a(6) = 3 since 6 = 1 + 2 + 3 = 2 + 1 + 3 = 3 + 2 + 1, and 2*1 + 1 = 3, 2*2 + 3 = 7, 2*3 + 5 = 11, 2*2 + 1 = 5, 2*1 + 3 = 5, 2*3 + 1 = 7, 2*1 + 5 = 7 are all prime.
a(10) = 1 since 10 = 3 + 4 + 3, and 2*3 + 1 = 7, 2*4 + 3 = 11, 2*3 + 5 = 11 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..6000
Zhi-Wei Sun, A new conjecture on triangular numbers, a message to Number Theory List, Oct. 25, 2013.
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Cf. A000040, A000217, A068307, A132399, A227877, A229166, A230121, A230141, A230219, A230596.

SQ[n_]:=IntegerQ[Sqrt[n]]
TQ[n_]:=SQ[8n+1]
a[n_]:=Sum[If[PrimeQ[2i+1]&&PrimeQ[2j+3]&&PrimeQ[2(n-i-j)+5]&&TQ[i*j(n-i-j)],1,0],{i,1,n-2},{j,1,n-1-i}]
Table[a[n],{n,1,100}]

cp's OEIS Frontend

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

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A230217 List of those primes p with p + 6 and 3*p + 8 also prime.

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A230502 Number of ways to write n = (2-(n mod 2))*p + q + r with p <= q <= r such that p, q, r, p^2 - 2, q^2 - 2, r^2 - 2 are all 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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A230493 Number of ways to write n = (2-(n mod 2))*p + q + r with p <= q <= r such that p, q, r, 2*p^2 - 1, 2*q^2 - 1, 2*r^2 - 1 are all prime.

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A236832 Number of ways to write 2*n - 1 = p + q + r (p <= q <= r) with p, q and r terms of A234695.

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

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

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

A230493 Number of ways to write n = (2-(n mod 2))p + q + r with p <= q <= r such that p, q, r, 2p^2 - 1, 2q^2 - 1, 2r^2 - 1 are all prime.

A230451 Number of ways to write n = x + y + z (x, y, z > 0) such that 2x + 1, 2y + 3, 2z + 5 are all prime and xy*z is a triangular number.