|
|
A134406
|
|
Composite numbers of the form k^2 + 1.
|
|
19
|
|
|
10, 26, 50, 65, 82, 122, 145, 170, 226, 290, 325, 362, 442, 485, 530, 626, 730, 785, 842, 901, 962, 1025, 1090, 1157, 1226, 1370, 1445, 1522, 1682, 1765, 1850, 1937, 2026, 2117, 2210, 2305, 2402, 2501, 2602, 2705, 2810, 3026, 3250, 3365, 3482, 3601, 3722, 3845
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
Square roots of these numbers are quadratic irrationals and corresponding chain fraction representations are periodic: sqrt(10) = [3;{2,3}], sqrt(26) = [5;{2,5}], sqrt(50) = [7;{2,7}], ..., where {} is denoted a period (we write {6} == {2,3}).
|
|
LINKS
|
|
|
FORMULA
|
|
|
EXAMPLE
|
10 is a term because 10 = 3^2 + 1 is composite,
26 is a term because 26 = 5^2 + 1 is composite,
50 is a term because 50 = 7^2 + 1 is composite.
|
|
MAPLE
|
ts_fn1:=proc(n) local i, tren, ans; ans:=[ ]: for i from 1 to n do tren := i^(2)+1: if (isprime(tren) = false) then ans:=[ op(ans), tren ]: fi od: RETURN(ans) end: ts_fn1(200);
|
|
MATHEMATICA
|
|
|
PROG
|
(Python)
from sympy import isprime
from itertools import count, takewhile
def aupto(limit):
form = takewhile(lambda x: x <= limit, (k**2+1 for k in count(1)))
return [number for number in form if not isprime(number)]
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|