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

1,1

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

Links

Table of n, a(n) for n=1..10.

Index entry for sequences related to reversing digits of squares

Example

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.

Prog

(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

Crossrefs

Cf. A027829, A033294.

Sequence in context: A071685 A008919 A110843 * A319570 A319482 A023093

Adjacent sequences: A354253 A354254 A354255 * A354257 A354258 A354259

Keyword

nonn,base,more

Author

Jon E. Schoenfield, May 21 2022

Extensions

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

Status

approved