OFFSET
1,1
COMMENTS
The sequence is infinite. The squares of the form 66...67^2 = 4..48..89 are terms.
Another infinite family is the squares 33...35^2 = 1...122...25. - Robert Israel, Aug 20 2019
REFERENCES
Ion Cucurezeanu, Perfect squares and cubes of integers, Ed. Gil, Zalău, (2007), ch. 4, p. 25, pr. 211, 212 (in Romanian).
LINKS
Chai Wah Wu, Table of n, a(n) for n = 1..286
EXAMPLE
5^2 = 25 = 2_(2 * 2 + 1);
7^2 = 49 = 4_(2 * 4 + 1);
35^2 = 1225 = 12_(2 * 12 + 1);
61907^2 = 3832476649 = 38324_(2 * 38324 + 1).
MAPLE
F:= proc(m) local x, X, A;
X:= [numtheory:-rootsunity(2, 10^m+2)];
A:= map(x -> (x^2-1)/(10^m+2), X);
A:= sort(select(x -> 2*x+1>=10^(m-1) and 2*x+1<10^m, A));
op(map(x -> x*10^m+2*x+1, A))
end proc:
subsop(1=NULL, [seq(F(m), m=1..10)]); # Robert Israel, Aug 20 2019
MATHEMATICA
Select[Array[FromDigits@ Flatten@ IntegerDigits[{#, 2 # + 1}] &, 10^5],
IntegerQ@ Sqrt@ # &] (* Michael De Vlieger, Aug 19 2019 *)
PROG
(Magma) [a:n in [1..30000000]|IsSquare(a) where a is 10^(#Intseq(2*n+1))*n+2*n+1];
(Python)
def Test(n):
s = str(n)
ps, ss = s[0:len(s)//2], s[len(s)//2:len(s)]
return int(ss) == 2*int(ps)+1 and s[len(s)//2] != "0"
n, a = 1, 4
while n < 23:
if Test(a*a):
print(n, a*a)
n = n+1
a = a+1 # A.H.M. Smeets, Aug 19 2019
(Python)
from itertools import count, islice
from sympy.ntheory.primetest import is_square
def A309828_gen(): # generator of terms
return filter(is_square, (int(str(k)+str((k<<1)+1)) for k in count(1)))
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Marius A. Burtea, Aug 18 2019
STATUS
approved