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A218825 Number of ways to write 2n-1 as p+2q with p, q and p^2+60q^2 all prime. 10
0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 1, 3, 1, 3, 3, 1, 2, 2, 1, 2, 3, 1, 2, 3, 1, 2, 2, 1, 3, 1, 1, 3, 3, 4, 3, 1, 2, 5, 3, 1, 3, 2, 4, 3, 3, 1, 7, 4, 1, 5, 3, 5, 8, 4, 3, 4, 3, 3, 5, 4, 4, 3, 2, 3, 5, 3, 5, 7, 3, 2, 9, 4, 4, 6, 3, 3, 8, 6, 1, 4, 5, 2, 7, 1, 4, 2, 4, 5, 5, 2, 4, 4, 3, 2, 5, 4, 5, 6, 4, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,9

COMMENTS

Conjecture: a(n)>0 for all n>8.

This conjecture is stronger than Lemoine's conjecture. It has been verified for n up to 10^8.

Zhi-Wei Sun also made the following general conjecture: For any positive integer n, the set E(n) of positive odd integers not of the form p+2q with p, q, p^2+4(2^n-1)q^2 all prime, is finite. In particular, if we let M(n) denote the maximal element of E(n), then M(1)=3449, M(2)=1711, E(3)={1,3,5,7,31,73}, E(4)={1,3,5,7,9,11,13,15},

  M(5)=6227, M(6)=1051, M(7)=2239, M(8)=2599, M(9)=7723,

  M(10)=781, M(11)=1163, M(12)=587, M(13)=11443,

  M(14)=2279, M(15)=157, M(16)=587, M(17)=32041,

  M(18)=1051, M(19)=2083, M(20)=4681.

Zhi-Wei Sun also guessed that for any positive even integer d not congruent to 2 modulo 6 there exists a prime p(d) such that for any prime p>p(d) there is a prime q<p with p^2+dq^2 prime. In particular, we may take

  p(4)=p(6)=3, p(10)=5, p(12)=3, p(16)=2, p(18)=3,

  p(22)=11, p(24)=17, and p(28)=p(30)=7.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..20000

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv preprint arXiv:1211.1588, 2012.

Wikipedia, Lemoine's conjecture

EXAMPLE

a(10)=1 since the only primes p and q with p^2+60q^2 prime and p+2q=19 are p=13 and q=3.

MATHEMATICA

a[n_]:=a[n]=Sum[If[PrimeQ[q]==True&&PrimeQ[2n-1-2q]==True&&PrimeQ[(2n-1-2q)^2+60q^2]==True, 1, 0], {q, 1, n-1}]

Do[Print[n, " ", a[n]], {n, 1, 20000}]

PROG

(PARI) A218825(n)={my(c=0, n21=n*2-1); forprime(q=2, n-1, isprime(n21-2*q) || next; isprime(q^2*60+(n21-2*q)^2) && c++); c}  \\ - M. F. Hasler, Nov 07 2012

CROSSREFS

Cf. A000040, A046927, A218754, A218585, A218654, A218656, A218797.

Sequence in context: A046111 A131779 A131775 * A035691 A131781 A082465

Adjacent sequences:  A218822 A218823 A218824 * A218826 A218827 A218828

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Nov 07 2012

STATUS

approved

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Last modified June 17 19:10 EDT 2019. Contains 324198 sequences. (Running on oeis4.)