|
%I #16 Aug 01 2021 13:01:57
%S 96,205,300,477,732,1920,3157,52896,120085,427020,8264460,88581312,
%T 112000885,112917765,143075580,152863360,193537077,233788192,
%U 266755221,313680096,370908477,386568925,440852992,442670220
%N Numbers m such that the concatenation of m with m+4 gives a square.
%C From _Farideh Firoozbakht_, Nov 26 2006: (Start)
%C 1. a(n).(a(n)+4) = A115438^2 where "." denotes concatenation.
%C 2. All numbers of the form f(j) = 4{j}.2.6{j-1}.70.2{j}.0 where each expression in braces denotes the multiplicity of the digit preceding the expression (e.g., "4{j}" means that the digit "4" appears j times consecutively) and where j > 0 are in the sequence because if k(j) = 6{j}.5.3{j}.4.6{j}.8 then k(j)^2 = f(j).(f(j)+4). For example, f(4) = 444426667022220, k(4) = 666653333466668, and k(4)^2 = 666653333466668^2 = f(4).(f(4)+4) = 444426667022220.444426667022224.
%C 3. All numbers of the form f(j) = 1{j}.2.0{j+1}.8{j}.5 where j > -1 are in the sequence because if k(j) = 3{j}.4.6{j}.5.3{j+1} then k(j)^2 = f(j).(f(j)+4). For example, f(5) = 111112000000888885, k(5) = 333334666665333333, and k(5)^2 = 333334666665333333^2 = f(5).(f(5)+4) = 111112000000888885.111112000000888889. (End)
%e Using "." to denote concatenation, 120085.120089 = 346533^2.
%t Select[Range[10^5],IntegerQ@Sqrt@FromDigits@Flatten[IntegerDigits/@{#,#+4}]&] (* _Giorgos Kalogeropoulos_, Jul 27 2021 *)
%Y Cf. A030465, A102567, A115426, A115428, A115429, A115430, A115431, A115432, A115433, A115434, A115435, A115436, A115438.
%K base,nonn
%O 1,1
%A _Giovanni Resta_, Jan 24 2006
|
