

A236166


Numbers with no nontrivial quadratic conjugates (see comments).


1



1, 2, 3, 4, 5, 7, 9, 13, 17, 25, 31, 67, 73, 313, 353, 361, 421, 541, 821, 947, 1033, 1123, 1321, 1453, 2371, 2381, 2707, 2887, 3041, 3319, 3461, 3481, 3727, 3767, 4021, 4201, 5521, 6701, 6947, 7757, 8011, 8087, 8971, 9811, 11027, 12157, 12289, 12547, 12577
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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 = (yx)^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 as1 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 aa2 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

Sequence in context: A064628 A188674 A320316 * A017834 A286225 A239048
Adjacent sequences: A236163 A236164 A236165 * A236167 A236168 A236169


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



