A343075 Digitally delicate square numbers (changing any one decimal digit always produces a nonsquare).

25, 121, 144, 169, 196, 256, 289, 324, 1024, 1089, 1156, 1296, 1369, 1444, 1521, 1681, 1764, 1849, 1936, 2500, 3136, 3249, 3364, 3481, 3721, 3844, 3969, 4096, 4356, 4489, 4624, 4761, 5041, 5184, 6084, 6241, 6561, 6724, 6889, 7056, 7396

Offset: 1

Author

Ctibor O. Zizka, Apr 04 2021

Keywords

• nonn
• base

Comments

If k is the count of digitally delicate square numbers <= n, then empirically lim_{n->oo} k/n = sqrt(5)/3.

Examples

			n = 25, changing the digit 2 in 25 to d5, d from {0,1,3,4,5,6,7,8,9} gives no square, changing the digit 5 in 25 to 2d, d from {0,1,2,3,4,6,7,8,9} gives no square. Thus n = 25 is a member of the sequence.
		

Crossrefs

Cf. A000290, A050249.

Programs

Mathematica

changes[n_] := Module[{d = IntegerDigits[n]}, FromDigits @ ReplacePart[d, First[#] -> Last[#]] & /@ Tuples[{Range[Length[d]], Range[0, 9]}]]; q[n_] := AllTrue[changes[n], # == n || ! IntegerQ @ Sqrt[#] &]; Select[Range[100]^2, q] (* Amiram Eldar, Apr 04 2021 *)

Python

from sympy import integer_nthroot
def is_square(n): return integer_nthroot(n, 2)[1]
def change1(n):
  s = str(n)
  for i in range(len(s)):
    for d in set("0123456789") - {s[i]}:
      yield int(s[:i] + d + s[i+1:])
def ok(sqr): return not any(is_square(t) for t in change1(sqr))
print(list(filter(ok, (k*k for k in range(87))))) # Michael S. Branicky, Apr 04 2021