A354256 Squares that remain square when written backward, are not divisible by 10, and have an even number of digits.

1089, 9801, 698896, 10036224, 42263001, 637832238736, 1021178969603881, 1883069698711201, 4099923883299904, 6916103777337773016196

Offset: 1

Author

Jon E. Schoenfield, May 21 2022

Keywords

• nonn
• base
• more

Comments

a(10) > 10^21.

Is this sequence infinite?

Every term is a multiple of 121.

Terms come in nonpalindromic pairs and palindromic singles; see Example section.

Removal of the "even number of digits" requirement yields A033294, which has 8560 terms < 10^20.

A027829 is a subsequence. - Chai Wah Wu, May 23 2022

Examples

			There are no 2-digit terms.
The smallest 4-digit multiple of 121 is 1089 = 33^2, which happens to be a(1); its digit reversal is a(2) = 9801 = 99^2.
The only 6-digit term is the palindrome a(3) = 698896 = 836^2.
The only 8-digit terms are a(4) = 10036224 = 3168^2 and its digit reversal a(5) = 42263001 = 6501^2.
There are no 10-digit terms.
The only 12-digit term is the palindrome a(6) = 637832238736 = 798644^2.
There are no 14-digit terms.
There are three 16-digit terms: a(7) = 1021178969603881 = 31955891^2, its digit reversal a(8) = 1883069698711201 = 43394351^2, and the palindrome a(9) = 4099923883299904 = 64030648^2.
		

Links

• Index entry for sequences related to reversing digits of squares

Crossrefs

Cf. A027829, A033294.

Programs

Mathematica

Select[Range[500000]^2,EvenQ[IntegerLength[#]]&&Mod[#,10]!=0&&IntegerQ[Sqrt[ IntegerReverse[ #]]]&] (* The program generates the first five terms of the sequence. *) (* Harvey P. Dale, Jul 28 2024 *)

Python

from math import isqrt
from itertools import count, islice
def sqr(n): return isqrt(n)**2 == n
def agen(): yield from (k*k for k in count(1) if k%10 and len(s:=str(k*k))%2==0 and sqr(int(s[::-1])))
print(list(islice(agen(), 6))) # Michael S. Branicky, May 23 2022

Python

from math import isqrt
from itertools import count, islice
from sympy import integer_nthroot
def A354256_gen(): # generator of terms
    for l in count(2,2):
        for m in (1,4,5,6,9):
            for k in range(1+isqrt(m*10**(l-1)-1),1+isqrt((m+1)*10**(l-1)-1)):
                if k % 10 and integer_nthroot(int(str(k*k)[::-1]),2)[1]:
                    yield k*k
A354256_list = list(islice(A354256_gen(),9)) # Chai Wah Wu, May 23 2022

Extensions

a(10) from Chai Wah Wu, May 24 2022