A230224 -id:A230224 - 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

A227908 Number of ways to write 2*n = p + q with p, q and (p-1)^2 + q^2 all prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Oct 12 2013

nonn

Conjecture: a(n) > 0 except for n = 1, 16, 292.
This implies not only Goldbach's conjecture for even numbers, but also Ming-Zhi Zhang's conjecture (cf. A036468) that any odd number greater than one can be written as x + y (x, y > 0) with x^2 + y^2 prime.
We have verified the conjecture for n up to 10^7.
Conjecture verified for n up tp 10^9. - Mauro Fiorentini, Sep 21 2023

			a(7) = 1 since 2*7 = 11 + 3, and (11-1)^2 + 3^2 = 109 is prime.
a(19) = 1 since 2*19 = 7 + 31, and (7-1)^2 + 31^2 = 997 is 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. A002375, A036468, A220554, A230224.

a[n_]:=Sum[If[PrimeQ[2n-Prime[i]]&&PrimeQ[(Prime[i]-1)^2+(2n-Prime[i])^2],1,0],{i,1,PrimePi[2n-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 *)

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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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A227908 Number of ways to write 2*n = p + q with p, q and (p-1)^2 + q^2 all prime.

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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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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.