A030666 Smallest nontrivial extension of n which is a square.

16, 25, 36, 49, 529, 64, 729, 81, 900, 100, 1156, 121, 1369, 144, 1521, 169, 1764, 1849, 196, 2025, 2116, 225, 2304, 2401, 256, 2601, 2704, 289, 2916, 3025, 3136, 324, 3364, 3481, 35344, 361, 3721, 3844, 3969, 400, 41209, 4225, 4356, 441

Offset: 1

Author

Patrick De Geest

Keywords

• nonn
• base

Comments

A trivial extension would mean appending no digits at all when n is already a square. With trivial extensions allowed, this sequence becomes A018796.

Examples

			80 is not a perfect square, but 81 = 9^2, so a(8) = 81.
Although 9 is already a square, we need to append at least one digit. However, none of 90, 91, 92, ..., 99 are squares. Then we try 900 = 30^2, so a(9) = 900.
		

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A030667, A030686, A018796.

See A264604 for another version (first differs at a(9)).

Programs

Maple

# Program which computes 20000 terms, from N. J. A. Sloane, Nov 24 2015
for b from 1 to 20000 do
sw1:=-1:
for p from 1 to 6 do
bp:=b*10^p;
for i from 0 to 10^p-1 do
if issqr(bp+i) then lprint(b,bp+i); sw1:=1; break; fi;
                       od:
if sw1 > 0 then break; fi;
                 od:
if sw1 < 0 then lprint("failed at b = ",b); fi;
                      od:

Python

from gmpy2 import isqrt
def A030666(n):
    d, nd = 10, 10*n
    while True:
        x = (isqrt(nd-1)+1)**2
        if x < nd+d:
            return int(x)
        d *= 10
        nd *= 10 # Chai Wah Wu, May 24 2016

Formula

a(n) = A030667(n)^2.