A287799 If x^2 + 2*y^2 is prime for all positive integers x and y with m = x*y then m is in the sequence.
1, 3, 21, 33, 123, 219, 321, 3453, 6621, 16521, 18273, 25089, 27831, 29787, 62313, 69981, 75459, 95577, 101301, 105459, 157299, 196239, 197481, 247047, 259797, 281433, 359943, 390237, 418881, 460821, 529167, 569559, 595869, 680307, 727341, 945141, 955569, 964401
Offset: 1
Keywords
Examples
1 = 1*1 and 1^2 + 2*1^2 = 3, a prime. 21 = 1*21 = 3*7 = 21*1 = 7*3 => 1^2 + 2*21^2 = 883, 3^2 + 2*7^2 = 107, 21^2 + 2*1^2 = 443 and 7^2 + 2*3^2 = 67 are primes.
Links
- Robert Israel, Table of n, a(n) for n = 1..300
- Wikipedia, Bunyakovsky conjecture.
Crossrefs
Programs
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Maple
filter:= proc(m) andmap(x -> isprime(x^2 + 2*(m/x)^2), numtheory:-divisors(m)); end proc: select(filter, [1, seq(seq(18*i+j, j=[3,15]),i=0..10^5)]); # Robert Israel, Jul 14 2017
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Mathematica
A287799 = {}; Do[ds = Divisors[n]; If[EvenQ[Length[ds]], flag = True; k = 1; While[k <= Length[ds]/2 && (criterion1 = PrimeQ[ds[[k]]^2 + 2 * ds[[-k]]^2]) && (criterion2 = PrimeQ[ds[[-k]]^2 + 2 * ds[[k]]^2]), k++]; If[criterion1 && criterion2, AppendTo[A287799, n]]], {n, 2, 10^6}]; A287799
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PARI
is(n) = d=divisors(n); for(i=1,#d, if(!isprime(d[i]^2 + 2*d[#d-i+1]^2), return(0))); n > 1 \\ David A. Corneth, Jun 01 2017
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Sage
R = range(1,100000) [m for m in R if all(is_prime(d^2+2*(m//d)^2) for d in divisors(m))] # Peter Luschny, Jun 18 2017
Extensions
Name reformulated and m=1 added by Wolfdieter Lang, Jun 20 2017
Comments