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URL: https://oeis.org/A309828

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A309828
Squares formed by concatenating k and 2*k+1.
2
25, 49, 1225, 4489, 112225, 444889, 11122225, 44448889, 816416329, 1111222225, 1451229025, 3832476649, 4444488889, 111112222225, 444444888889, 10185602037121, 11111122222225, 44444448888889, 46355849271169, 997230019944601, 1111111222222225, 1231148024622961
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
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)))
A309828_list = list(islice(A309828_gen(), 20)) # Chai Wah Wu, Feb 20 2023
KEYWORD
nonn,base
AUTHOR
Marius A. Burtea, Aug 18 2019
STATUS
approved