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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. Conjecture verified for 2n-1 up to 10^9. - Mauro Fiorentini, Jul 20 2023 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. Conjecture verified for 2n-1 up to 10^9 for n <= 4 and up to 10^6 for n <= 20. - Mauro Fiorentini, Jul 20 2023 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

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Last modified December 8 06:27 EST 2023. Contains 367662 sequences. (Running on oeis4.)