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A287930 Numbers m such that for any positive integers (x, y), if x * y = m where x <= y, then x^2 + 2*y^2 is a prime number. 1
1, 3, 21, 33, 93, 105, 123, 177, 219, 237, 321, 417, 489, 537, 633, 699, 813, 951, 1011, 1299, 1419, 1641, 1923, 1959, 2073, 2211, 2433, 2661, 3387, 3453, 3489, 3741, 3981, 4083, 4377, 4461, 4467, 4827, 4911, 5007, 5997, 6423, 6621, 7467, 7647, 7881, 8031, 8061 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The sequence contains A287799.

a(n) == 3 or 15 (mod 18) for n > 1.

The numbers a(n)/3 are 1, 7, 11, 31, 35, 41, 59, 73, 79, 107, ... with a majority of prime numbers, except the subset {b(m)} = {1, 35, 473, 737, 1247, 2489, 2627, ...}. It seems that b(m) is semiprime for m > 1.

From Robert Israel, Jul 13 2017: (Start)

Not all b(m) for m > 1 are semiprime.

A counterexample is a(8821) = 23963385 = 3*5*373*4283.

All terms are squarefree. (End)

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

EXAMPLE

105 = 1*105 = 3*35 = 5*21 = 7*15 => 1^2 + 2*105^2 = 22051, 3^2 + 2*35^2 = 2459, 5^2 + 2*21^2 = 907 and 7^2 + 2*15^2 = 499 are primes.

MAPLE

filter:= proc(m)

   andmap(x -> isprime(x^2 + 2*(m/x)^2),

     select(t -> t^2 <= m, numtheory:-divisors(m)));

end proc:

select(filter, [1, seq(i, i=3..10000, 3)]); # Robert Israel, Jul 13 2017

MATHEMATICA

t={}; Do[ds=Divisors[n]; If[EvenQ[Length[ds]], ok=True; k=1; While[k<=Length[ds]/2&&(ok=PrimeQ[ds[[k]]^2+2*ds[[-k]]^2]), k++]; If[ok, AppendTo[t, n]]], {n, 2, 10^4}]; t

CROSSREFS

Cf. A000040, A287799.

Sequence in context: A317860 A039766 A072849 * A287799 A089323 A100986

Adjacent sequences:  A287927 A287928 A287929 * A287931 A287932 A287933

KEYWORD

nonn

AUTHOR

Michel Lagneau, Jun 03 2017

EXTENSIONS

Edited by Robert Israel, Jul 13 2017

STATUS

approved

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Last modified April 11 14:44 EDT 2021. Contains 342886 sequences. (Running on oeis4.)