A034822 Numbers k such that there are no palindromic squares of length k.

2, 4, 8, 10, 14, 18, 20, 24, 30, 38, 40, 46, 54, 56, 62, 64

Offset

1,1

Comments

All terms are even since (10^k+1)^2 is a palindrome of length 2*k+1. - Chai Wah Wu, Jun 14 2024

References

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 279.

Links

Table of n, a(n) for n=1..16.

Patrick De Geest, Palindromic Squares in bases 2 to 17

Eric Weisstein's World of Mathematics, Palindromic Number

Mathematica

A034822[n_] := Select[Range[Ceiling[Sqrt[10^(n - 1)]], Floor[Sqrt[10^n]]], #^2 == IntegerReverse[#^2] &];
Select[Range[12], Length[A034822[#]] == 0 &] (* Robert Price, Apr 23 2019 *)

Prog

(Python)
from sympy import integer_nthroot as iroot
def ispal(n): s = str(n); return s == s[::-1]
def ok(n):
  for r in range(iroot(10**(n-1), 2)[0] + 1, iroot(10**n, 2)[0]):
    if ispal(r*r): return False
  return True
print([m for m in range(1, 16) if ok(m)]) # Michael S. Branicky, Feb 04 2021

Crossrefs

Cf. A002778, A002779, A034307, A263618.

Sequence in context: A189670 A341822 A363797 * A300781 A050567 A069879

Adjacent sequences: A034819 A034820 A034821 * A034823 A034824 A034825

Keyword

nonn,base,hard,more

Author

Patrick De Geest, Oct 15 1998

Extensions

Two more terms from Patrick De Geest, Apr 01 2002

a(12)-a(16) from A263618 added by Max Alekseyev, Jun 03 2026

Status

approved