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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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified November 24 18:11 EST 2020. Contains 338616 sequences. (Running on oeis4.)