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A236166 Numbers with no nontrivial quadratic conjugates (see comments). 1

%I

%S 1,2,3,4,5,7,9,13,17,25,31,67,73,313,353,361,421,541,821,947,1033,

%T 1123,1321,1453,2371,2381,2707,2887,3041,3319,3461,3481,3727,3767,

%U 4021,4201,5521,6701,6947,7757,8011,8087,8971,9811,11027,12157,12289,12547,12577

%N Numbers with no nontrivial quadratic conjugates (see comments).

%C We call positive integers x and y quadratically conjugated by r if the x^2 mod y = y^2 mod x = r.

%C The following cases of quadratic conjugation are trivial:

%C 1) r = 0;

%C 2) r = (y-x)^2;

%C 3) y = x^2 - 1 (where x < y).

%C Every positive integer x has trivial quadratically conjugated numbers.

%C Every positive integer x has only a finite number of quadratically conjugated numbers, of which y = x^2 is the largest.

%C All known terms (except 1) are primes or squares of primes.

%C The largest known composite terms are 197^2 and 641^2.

%H Jon E. Schoenfield, <a href="/A236166/b236166.txt">Table of n, a(n) for n = 1..75</a>

%e 9 is in the sequence because all quadratically conjugated numbers to 9 are trivial: 3, 7, 8, 9, 10, 11, 27, 80, 81.

%e 6 is not in the sequence because it has the nontrivial quadratically conjugated number 16: 16^2 mod 6 = 6^2 mod 16 = 4.

%p coq0:=proc(a:=posint) local s,aa,b,c,f;

%p f:=true:s:=floor(sqrt(a)):aa:=a^2:

%p for b from 2 to a-s-1 do c:=b^2 mod a:

%p if c>0 then if aa mod b = c then f:=false:break fi fi od:

%p if f then for b from a+s+1 to aa-2 do c:=b^2 mod a:

%p if c>0 then if aa mod b = c then f:=false:break fi fi od fi:

%p f end:

%p A:={}:for i to 2000 do if coq0(i) then A:=A union {i}: print(ifactor(i)) fi od:

%p # or

%p for i to 2000 do if coq0(i) then print(i) fi od:

%p # _Vladimir Letsko_, Dec 17 2014

%K nonn

%O 1,2

%A _Vladimir Letsko_, Jan 19 2014

%E Missing term a(10)=25 inserted and terms a(45)-a(49) added by _Jon E. Schoenfield_, Aug 16 2015

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Last modified January 16 01:34 EST 2021. Contains 340195 sequences. (Running on oeis4.)