A243091 Least number k > n such that n concatenated with k is a perfect square.

1, 6, 5, 6, 9, 29, 25, 29, 41, 61, 24, 56, 25, 69, 44, 21, 81, 64, 49, 36, 25, 316, 201, 104, 336, 281, 244, 225, 224, 241, 276, 36, 49, 64, 81, 344, 100, 249, 44, 69, 96, 209, 436, 56, 89, 369, 225, 61, 400, 284, 176, 84, 441, 361, 76, 225, 169, 76, 564, 536, 84, 504, 500, 504, 516, 536

Offset: 0

Author

Derek Orr, Aug 18 2014

Keywords

• nonn
• base

Comments

Records occur at: 0, 1, 4, 5, 8, 9, 13, 16, 21, 24, 35, 42, 52, 58, 67, 75, 80, ..., . - Robert G. Wilson v, Nov 23 2015

Examples

			a(1) = 6 since 6>1 and 16 = 4^2.
a(2) = 5 since 5>2 and 25 = 5^2.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A090566.

Programs

Mathematica

f[n_] := Block[{x = n, d = 1 + Floor@ Log10@ n}, q = (Floor@ Sqrt[(10^d + 1) x] + 1)^2; If[q < (10^d) (x + 1), Mod[q, 10^d], Mod[(Floor@ Sqrt[(10^d) (10 x + 1) - 1] + 1)^2, 10^(d + 1)]]]; Array[f, 65] (* Robert G. Wilson v, Nov 23 2015, after the algorithm of David W. Wilson in A090566 *)
lnk[n_]:=Module[{k=n+1},While[!IntegerQ[Sqrt[n 10^IntegerLength[k]+k]],k++];k]; Array[lnk,70,0] (* Harvey P. Dale, Sep 01 2023 *)

PARI

a(n)=s=Str(n); k=n+1; while(!issquare(eval(concat(s,Str(k)))), k++); return(k)
vector(100, n, a(n))

PARI

A048761 = t->(sqrtint(t-1)+1)^2
A243091(n)={my(d=#Str(n),a=A048761((1+10^d)*n)); a>=(n+1)*10^d && a=A048761((n*10+1)*10^d); a%10^(d+(a>=100^d))} \\ M. F. Hasler, Nov 24 2015

Extensions

a(0)=1 added by N. J. A. Sloane, Nov 24 2015