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

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

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

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