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A218825
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Number of ways to write 2n-1 as p+2q with p, q and p^2+60q^2 all prime.
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10
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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
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OFFSET
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1,9
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COMMENTS
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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.
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<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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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}]
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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