A230261 Number of ways to write 2*n - 1 = p + q with p, p + 6 and q^4 + 1 all prime, where q is a positive integer.

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

Offset

1,5

Comments

Conjecture: (i) a(n) > 0 for all n > 3. Also, any odd number greater than 6 can be written as p + q (q > 0) with p, p + 6 and q^2 + 1 all prime.

(ii) Any integer n > 1 can be written as x + y (x, y > 0) with x^4 + 1 and y^2 + y + 1 both prime.

(iii) Each integer n > 2 can be expressed as x + y (x, y > 0) with 4*x^2 + 3 and 4*y^2 -3 both prime.

Either of parts (i) and (ii) implies that there are infinitely many primes of the form x^4 + 1.

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(6) = 2 since 2*6-1 = 5 + 6 = 7 + 4, and 5, 5+6 = 11, 7, 7+6 = 13, 6^4+1 = 1297 and 4^4+1 = 257 are all prime.
a(25) = 1 since 2*25-1 = 47 + 2, and 47, 47+6 = 53, 2^4+1 = 17 are all prime.

Mathematica

a[n_]:=Sum[If[PrimeQ[Prime[i]+6]&&PrimeQ[(2n-1-Prime[i])^4+1], 1, 0], {i, 1, PrimePi[2n-2]}]
Table[a[n], {n, 1, 100}]

Prog

(PARI) a(n)=my(s, p=5, q=7); forprime(r=11, 2*n+4, if(r-p==6&&isprime((2*n-1-p)^4+1), s++); if(r-q==6&&isprime((2*n-1-q)^4+1), s++); p=q; q=r); s \\ Charles R Greathouse IV, Oct 14 2013

Crossrefs

Cf. A000068, A023201, A037896, A219864, A227908, A227909, A230241, A230252, A230254.

Sequence in context: A191291 A131841 A309432 * A242306 A245636 A171622

Adjacent sequences: A230258 A230259 A230260 * A230262 A230263 A230264

Keyword

nonn

Author

Zhi-Wei Sun, Oct 14 2013

Status

approved