A071176 Smallest k such that the concatenation of n and k is a square (decimal notation).

6, 5, 6, 9, 29, 4, 29, 1, 61, 0, 56, 1, 69, 4, 21, 9, 64, 49, 6, 25, 16, 5, 104, 336, 6, 244, 225, 9, 16, 25, 36, 4, 64, 81, 344, 1, 21, 44, 69, 0, 209, 25, 56, 1, 369, 24, 61, 4, 284, 41, 84, 9, 29, 76, 225, 25, 6, 564, 29, 84, 504, 5, 504

Offset: 1

Author

Reinhard Zumkeller, May 15 2002

Keywords

• nonn
• base
• nice
• look

Comments

a(n) = 1 correspond to n = A132356(m), m > 0. - Bill McEachen, Aug 31 2023

Examples

			a(5) = 29 as 529 = 23^2 and 5'i is nonsquare for i<29, A071177(5)=23.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000196, A071177, A245631, A132356.

Programs

Haskell

import Data.List (findIndex)
import Data.Maybe (fromJust)
a071176 n = fromJust $ findIndex (== 1) $
            map (a010052 . read . (show n ++) . show) [0..]
-- Reinhard Zumkeller, Aug 09 2011

Mathematica

nksq[n_]:=Module[{idn=IntegerDigits[n],k=0},While[!IntegerQ[Sqrt[ FromDigits[Join[ idn,IntegerDigits[k]]]]],k++];k]; Array[nksq,70] (* Harvey P. Dale, Sep 28 2012 *)

PARI

a(n)={if(issquare(10*n), 0, my(m=n, b=1); while(1, m*=10; my(r=(sqrtint(m+b-1)+1)^2-m); b*=10; if(rAndrew Howroyd, Jan 13 2023

Python

from math import isqrt
from sympy.ntheory.primetest import is_square
def A071176(n):
    m = 10*n
    if is_square(m): return 0
    a = 1
    while (k:=(isqrt(a*(m+1)-1)+1)**2-m*a)>=10*a:
        a *= 10
    return k # Chai Wah Wu, Feb 15 2023

Formula

A000196(n . a(n)) = A071177(n) where "." stands for concatenation.