A068165 Smallest square formed from n by inserting zero or more decimal digits.

1, 25, 36, 4, 25, 16, 576, 81, 9, 100, 121, 121, 1369, 144, 1156, 16, 1764, 1089, 169, 2025, 121, 225, 2304, 324, 25, 256, 2704, 289, 289, 2304, 361, 324, 3136, 324, 3025, 36, 3721, 3481, 1369, 400, 441, 4225, 4356, 144, 4225, 4096, 4761, 484, 49, 2500

Offset: 1

Author

Amarnath Murthy, Feb 25 2002

Keywords

• base
• easy
• nonn

Comments

The digits may be added before, in the middle of, or after the digits of n.

If n is a square then a(n) = n. - Zak Seidov, Nov 13 2014

Examples

			Smallest square formed from 20 is 2025, by placing 25 on the right side. Smallest square formed from 33 is 3136, by inserting a 1 between 3's and placing a 6.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000

Crossrefs

A091873 gives square roots. Cf. A038690, A029944, A068164.

Programs

Maple

A068165 := proc(n)
        local b,pdigs,plen,dmas,dmasdigs,i,j;
        for b from 1 do
                pdigs := convert(b^2,base,10) ;
                plen := nops(pdigs) ;
                for dmas from 2^plen-1 to 0 by -1 do
                        dmasdigs := convert(dmas,base,2) ;
                        pdel := [] ;
                        for i from 1 to nops(dmasdigs) do
                                if op(i,dmasdigs) = 1 then
                                        pdel := [op(pdel),op(i,pdigs)] ;
                                end if;
                        end do:
                        if n = add(op(j,pdel)*10^(j-1),j=1..nops(pdel)) then
                                return b^2;
                        end if;
                end do:
        end do:
end proc:
seq(A068165(n),n=1..133) ; # R. J. Mathar, Nov 14 2014

Python

from math import isqrt
from itertools import count
def dmo(n, t):
    if t < n: return False
    while n and t:
        if n%10 == t%10:
            n //= 10
        t //= 10
    return n == 0
def a(n):
    return next(t for t in (i*i for i in count(isqrt(n))) if dmo(n, t))
print([a(n) for n in range(1, 77)]) # Michael S. Branicky, Jan 21 2023

Extensions

Corrected and extended by Ray Chandler, Oct 11 2003