A115437 Numbers m such that the concatenation of m with m+4 gives a square.

96, 205, 300, 477, 732, 1920, 3157, 52896, 120085, 427020, 8264460, 88581312, 112000885, 112917765, 143075580, 152863360, 193537077, 233788192, 266755221, 313680096, 370908477, 386568925, 440852992, 442670220

Offset

1,1

Comments

From Farideh Firoozbakht, Nov 26 2006: (Start)

1. a(n).(a(n)+4) = A115438^2 where "." denotes concatenation.

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.

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)

Links

Table of n, a(n) for n=1..24.

Example

Using "." to denote concatenation, 120085.120089 = 346533^2.

Mathematica

Select[Range[10^5], IntegerQ@Sqrt@FromDigits@Flatten[IntegerDigits/@{#, #+4}]&] (* Giorgos Kalogeropoulos, Jul 27 2021 *)

Crossrefs

Cf. A030465, A102567, A115426, A115428, A115429, A115430, A115431, A115432, A115433, A115434, A115435, A115436, A115438.

Sequence in context: A352975 A044428 A044809 * A347746 A241930 A270435

Adjacent sequences: A115434 A115435 A115436 * A115438 A115439 A115440

Keyword

base,nonn

Author

Giovanni Resta, Jan 24 2006

Status

approved