A227898 -id:A227898 - cp's OEIS frontend

A227899 Number of primes p < n with 3*p - 4 and n^2 + (n - p)^2 both prime.

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

Offset: 1

Zhi-Wei Sun, Oct 14 2013

nonn

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

			a(5) = 1 since 5 = 3 + 2, and the three numbers 3, 3*3 - 4 = 5 and 5^2 + (5-3)^2 = 29 are 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.

Cf. A069003, A185636, A204065, A227898, A230223.

a[n_]:=Sum[If[PrimeQ[3Prime[i]-4]&&PrimeQ[n^2+(n-Prime[i])^2],1,0],{i,1,PrimePi[n-1]}]
Table[a[n],{n,1,100}]

A232194 Number of ways to write n = x + y (x, y > 0) with nx + y and ny - x both prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Nov 20 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 2. Also, a(n) = 1 only for n = 3, 4, 6, 20, 24.
(ii) Any positive integer n not among 1, 30, 54 can be written as x + y (x, y > 0) with n*x + y and n*y + x both prime.
(iii) Each integer n > 1 not equal to 8 can be expressed as x + y (x, y > 0) with n*x^2 + y (or x^4 + n*y) prime.
(iv) Any integer n > 5 can be written as p + q (q > 0) with p and n*q^2 + 1 both prime.
See also A232174 for a similar conjecture.

			a(3) = 1 since 3 = 1 + 2 with 3*1 + 2 = 3*2 - 1 = 5 prime.
a(4) = 1 since 4 = 1 + 3 with 4*1 + 3 = 7 and 4*3 - 1 = 11 both prime.
a(6) = 1 since 6 = 1 + 5 with 6*1 + 5 = 11 and 6*5 - 1 = 29 both prime.
a(20) = 1 since 20 = 9 + 11 with 20*9 + 11 = 191 and 20*11 - 9 = 211 both prime.
a(24) = 1 since 24*19 + 5 = 461 and 24*5 - 19 = 101 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.

Cf. A000040, A218585, A218654, A219864, A220413, A227898, A227899, A232174, A232186.

a[n_]:=Sum[If[PrimeQ[n*x+(n-x)]&&PrimeQ[n*(n-x)-x],1,0],{x,1,n-1}]
Table[a[n],{n,1,100}]

A232109 Least prime p < n + 5 with n + (p-1)*(p-3)/8 prime, or 0 if such a prime p does not exist.

Original entry on oeis.org

5, 3, 3, 5, 3, 5, 3, 7, 11, 5, 3, 5, 3, 7, 17, 5, 3, 5, 3, 7, 11, 5, 3, 23, 17, 7, 11, 5, 3, 5, 3, 13, 11, 7, 19, 5, 3, 7, 17, 5, 3, 5, 3, 7, 17, 5, 3, 23, 11, 7, 11, 5, 3, 23, 17, 7, 11, 5, 3, 5, 3, 31, 11, 7, 19, 5, 3, 7, 11, 5, 3, 5, 3, 13, 17, 7, 19, 5, 3, 7, 17, 5, 3, 23, 17, 7, 11, 5, 3, 29, 11, 13, 11, 7, 19, 5, 3, 7, 11, 5

Offset: 1

Zhi-Wei Sun, Nov 18 2013

nonn

Conjecture: a(n) > 0 for all n > 0. Moreover, for any integer n > 1 there exists a prime p < 2*sqrt(n)*log(7n) such that n + (p-1)*(p-3)/8 is prime.

This implies that any integer n > 1 can be written as (p-1)/2 + q with q a positive integer, and p and (p^2-1)/8 + q both prime.

			a(1) = 5 since neither 1 + (2-1)*(2-3)/8 = 7/8 nor 1 + (3-1)*(3-3)/8 = 1  is prime, but 1 + (5-1)*(5-3)/8 = 2 is prime.

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

Cf. A000040, A185636, A204065, A227898, A228425, A231201, A231557.

Do[Do[If[PrimeQ[n+(Prime[k]-1)(Prime[k]-3)/8],Goto[aa]],{k,1,PrimePi[n+4]}];
Print[n," ",0];Label[aa];Continue,{n,1,100}]

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A227899 Number of primes p < n with 3*p - 4 and n^2 + (n - p)^2 both prime.

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A232194 Number of ways to write n = x + y (x, y > 0) with nx + y and ny - x both prime.

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A232109 Least prime p < n + 5 with n + (p-1)*(p-3)/8 prime, or 0 if such a prime p does not exist.

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cp's OEIS Frontend

A227899 Number of primes p < n with 3*p - 4 and n^2 + (n - p)^2 both prime.

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A232194 Number of ways to write n = x + y (x, y > 0) with n*x + y and n*y - x both prime.

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Mathematica

A232109 Least prime p < n + 5 with n + (p-1)*(p-3)/8 prime, or 0 if such a prime p does not exist.

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Comments

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Mathematica

A232194 Number of ways to write n = x + y (x, y > 0) with nx + y and ny - x both prime.