A115439 - OEIS

%I #25 Jul 09 2023 11:21:14

%S 4,7,45,56,38163,61838,83618,346980,653021,950051,8647555,9534265,

%T 8167822283,9007920992,9209900792,9950000501,4737445289221,

%U 4990568257187,5009431742814,5262554710780,8373808925585,8626931893551,34323166122692,34532758615690,49625657225895,49835249718893

%N Numbers m such that the square of m is the concatenation of two numbers k and k+5.

%C All numbers of the form f(n)=9(n).5.0(2n).5.0(n-1).1 where n>0 are in the sequence because if k(n)=9(n).0(n).25.0(n-1).9(n).6 then f(n)^2=k(n).(k(n)+5). For example f(2)=9950000501; k(2)=9900250996 and f(2)^2=9950000501^2=9900250996.9900251001 =k(2).(k(2)+5). - _Farideh Firoozbakht_, Nov 26 2006

%C m^2 = (k)|(k+5) = (k)|(k) + 5 = (10^q + 1)*k + 5 where | denotes concatenation and q is the number of digits of k gives a nonlinear equation that can be solved using the solver below. - _David A. Corneth_, Jan 02 2021

%H David A. Corneth, <a href="/A115439/b115439.txt">Table of n, a(n) for n = 1..3733</a>

%H Dario A. Alpern, <a href="https://www.alpertron.com.ar/QUAD.HTM">Generic two integer variable equation solver</a>

%e 38163^2 = 14564_14569.

%Y Cf. A030467, A057934, A106497, A115428, A115427, A115438, A115440, A115441, A115442, A115443, A115444, A115445, A115446, A115447.

%K base,nonn

%O 1,1

%A _Giovanni Resta_, Jan 24 2006

%E More terms from _David A. Corneth_, Jan 02 2021