OFFSET
1,2
COMMENTS
Convenient numbers (A000926) are numbers n such that for all primes p where p and p-n are quadratic residues (mod 4*n), p can be written as x^2 + n*y^2, so convenient numbers are a subsequence.
All non-convenient numbers which are members of this sequence are either congruent to 0 (mod 4) or 3 (mod 4).
The equation 4*p = x^2 + n*y^2 is important because if n is a squarefree integer congruent to 3 (mod 4), then the ring of integers Q[sqrt(-n)] will be all integers of form (x/2) + (y/2)*sqrt(-n) for x and y of the same parity, whose norm is (x/2)^2 + n*(y/2)^2. If prime p = (x/2)^2 + n*(y/2)^2, then 4*p = x^2 + n*y^2.
Is this sequence finite?
Is 7392 the largest term of this sequence?
There are no further terms up to 10^6. - Andrew Howroyd, Jun 08 2018
LINKS
V. Raman, Table of n, a(n) for n = 1..139
Thomas R. Hagedorn, Primes of the form x^2+ny^2 and the geometry of (convenient) numbers
EXAMPLE
n = 14 is not a member of this sequence because for prime p = 71, 4*p = 284 cannot be written as x^2 + 14*y^2.
PROG
(PARI) ok(n)=!#select(k->k<>2, quadclassunit(-n*if((-n)%4>1, 4, 1)).cyc) \\ Andrew Howroyd, Jun 08 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
V. Raman, Nov 25 2013
STATUS
approved