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

%I #52 Mar 18 2023 16:27:32

%S 25,49,1225,4489,112225,444889,11122225,44448889,816416329,1111222225,

%T 1451229025,3832476649,4444488889,111112222225,444444888889,

%U 10185602037121,11111122222225,44444448888889,46355849271169,997230019944601,1111111222222225,1231148024622961

%N Squares formed by concatenating k and 2*k+1.

%C The sequence is infinite. The squares of the form 66...67^2 = 4..48..89 are terms.

%C Another infinite family is the squares 33...35^2 = 1...122...25. - _Robert Israel_, Aug 20 2019

%D Ion Cucurezeanu, Perfect squares and cubes of integers, Ed. Gil, Zalău, (2007), ch. 4, p. 25, pr. 211, 212 (in Romanian).

%H Chai Wah Wu, <a href="/A309828/b309828.txt">Table of n, a(n) for n = 1..286</a>

%e 5^2 = 25 = 2_(2 * 2 + 1);

%e 7^2 = 49 = 4_(2 * 4 + 1);

%e 35^2 = 1225 = 12_(2 * 12 + 1);

%e 61907^2 = 3832476649 = 38324_(2 * 38324 + 1).

%p F:= proc(m) local x,X,A;

%p X:= [numtheory:-rootsunity(2,10^m+2)];

%p A:= map(x -> (x^2-1)/(10^m+2), X);

%p A:= sort(select(x -> 2*x+1>=10^(m-1) and 2*x+1<10^m, A));

%p op(map(x -> x*10^m+2*x+1, A))

%p end proc:

%p subsop(1=NULL, [seq(F(m),m=1..10)]); # _Robert Israel_, Aug 20 2019

%t Select[Array[FromDigits@ Flatten@ IntegerDigits[{#, 2 # + 1}] &, 10^5],

%t IntegerQ@ Sqrt@ # &] (* _Michael De Vlieger_, Aug 19 2019 *)

%o (Magma) [a:n in [1..30000000]|IsSquare(a) where a is 10^(#Intseq(2*n+1))*n+2*n+1];

%o (Python)

%o def Test(n):

%o s = str(n)

%o ps, ss = s[0:len(s)//2], s[len(s)//2:len(s)]

%o return int(ss) == 2*int(ps)+1 and s[len(s)//2] != "0"

%o n, a = 1, 4

%o while n < 23:

%o if Test(a*a):

%o print(n,a*a)

%o n = n+1

%o a = a+1 # _A.H.M. Smeets_, Aug 19 2019

%o (Python)

%o from itertools import count, islice

%o from sympy.ntheory.primetest import is_square

%o def A309828_gen(): # generator of terms

%o return filter(is_square,(int(str(k)+str((k<<1)+1)) for k in count(1)))

%o A309828_list = list(islice(A309828_gen(),20)) # _Chai Wah Wu_, Feb 20 2023

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

%K nonn,base

%O 1,1

%A _Marius A. Burtea_, Aug 18 2019