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

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

1,5

Comments

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.

Links

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.

Example

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.

Mathematica

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}]

Crossrefs

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

Sequence in context: A053276 A064065 A284486 * A054705 A025800 A029258

Adjacent sequences: A232191 A232192 A232193 * A232195 A232196 A232197

Keyword

nonn

Author

Zhi-Wei Sun, Nov 20 2013

Status

approved