A309828 Squares formed by concatenating k and 2*k+1.

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

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)))
A309828_list = list(islice(A309828_gen(), 20)) # Chai Wah Wu, Feb 20 2023

Crossrefs

Cf. A000290, A030466, A054215, A109344, A181719, A309808, A309809.

Sequence in context: A056981 A132585 A049228 * A076637 A040600 A277190

Adjacent sequences: A309825 A309826 A309827 * A309829 A309830 A309831

Keyword

nonn,base

Author

Marius A. Burtea, Aug 18 2019

Status

approved