OFFSET
1,2
COMMENTS
We call positive integers x and y quadratically conjugated by r if the x^2 mod y = y^2 mod x = r.
The following cases of quadratic conjugation are trivial:
1) r = 0;
2) r = (y-x)^2;
3) y = x^2 - 1 (where x < y).
Every positive integer x has trivial quadratically conjugated numbers.
Every positive integer x has only a finite number of quadratically conjugated numbers, of which y = x^2 is the largest.
All known terms (except 1) are primes or squares of primes.
The largest known composite terms are 197^2 and 641^2.
LINKS
Jon E. Schoenfield, Table of n, a(n) for n = 1..75
EXAMPLE
9 is in the sequence because all quadratically conjugated numbers to 9 are trivial: 3, 7, 8, 9, 10, 11, 27, 80, 81.
6 is not in the sequence because it has the nontrivial quadratically conjugated number 16: 16^2 mod 6 = 6^2 mod 16 = 4.
MAPLE
coq0:=proc(a:=posint) local s, aa, b, c, f;
f:=true:s:=floor(sqrt(a)):aa:=a^2:
for b from 2 to a-s-1 do c:=b^2 mod a:
if c>0 then if aa mod b = c then f:=false:break fi fi od:
if f then for b from a+s+1 to aa-2 do c:=b^2 mod a:
if c>0 then if aa mod b = c then f:=false:break fi fi od fi:
f end:
A:={}:for i to 2000 do if coq0(i) then A:=A union {i}: print(ifactor(i)) fi od:
# or
for i to 2000 do if coq0(i) then print(i) fi od:
# Vladimir Letsko, Dec 17 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Letsko, Jan 19 2014
EXTENSIONS
Missing term a(10)=25 inserted and terms a(45)-a(49) added by Jon E. Schoenfield, Aug 16 2015
STATUS
approved