

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.


3



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
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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

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.


EXAMPLE

a(6) = 2 since 2*61 = 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*251 = 47 + 2, and 47, 47+6 = 53, 2^4+1 = 17 are all prime.


MATHEMATICA

a[n_]:=Sum[If[PrimeQ[Prime[i]+6]&&PrimeQ[(2n1Prime[i])^4+1], 1, 0], {i, 1, PrimePi[2n2]}]
Table[a[n], {n, 1, 100}]


PROG

(PARI) a(n)=my(s, p=5, q=7); forprime(r=11, 2*n+4, if(rp==6&&isprime((2*n1p)^4+1), s++); if(rq==6&&isprime((2*n1q)^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

ZhiWei Sun, Oct 14 2013


STATUS

approved



