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

%I A354256 #20 Jul 28 2024 11:45:40
%S A354256 1089,9801,698896,10036224,42263001,637832238736,1021178969603881,
%T A354256 1883069698711201,4099923883299904,6916103777337773016196
%N A354256 Squares that remain square when written backward, are not divisible by 10, and have an even number of digits.
%C A354256 a(10) > 10^21.
%C A354256 Is this sequence infinite?
%C A354256 Every term is a multiple of 121.
%C A354256 Terms come in nonpalindromic pairs and palindromic singles; see Example section.
%C A354256 Removal of the "even number of digits" requirement yields A033294, which has 8560 terms < 10^20.
%C A354256 A027829 is a subsequence. - _Chai Wah Wu_, May 23 2022
%H A354256 <a href="/index/Sq#sqrev">Index entry for sequences related to reversing digits of squares</a>
%e A354256 There are no 2-digit terms.
%e A354256 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.
%e A354256 The only 6-digit term is the palindrome a(3) = 698896 = 836^2.
%e A354256 The only 8-digit terms are a(4) = 10036224 = 3168^2 and its digit reversal a(5) = 42263001 = 6501^2.
%e A354256 There are no 10-digit terms.
%e A354256 The only 12-digit term is the palindrome a(6) = 637832238736 = 798644^2.
%e A354256 There are no 14-digit terms.
%e A354256 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.
%t A354256 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 *)
%o A354256 (Python)
%o A354256 from math import isqrt
%o A354256 from itertools import count, islice
%o A354256 def sqr(n): return isqrt(n)**2 == n
%o A354256 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])))
%o A354256 print(list(islice(agen(), 6))) # _Michael S. Branicky_, May 23 2022
%o A354256 (Python)
%o A354256 from math import isqrt
%o A354256 from itertools import count, islice
%o A354256 from sympy import integer_nthroot
%o A354256 def A354256_gen(): # generator of terms
%o A354256     for l in count(2,2):
%o A354256         for m in (1,4,5,6,9):
%o A354256             for k in range(1+isqrt(m*10**(l-1)-1),1+isqrt((m+1)*10**(l-1)-1)):
%o A354256                 if k % 10 and integer_nthroot(int(str(k*k)[::-1]),2)[1]:
%o A354256                     yield k*k
%o A354256 A354256_list = list(islice(A354256_gen(),9)) # _Chai Wah Wu_, May 23 2022
%Y A354256 Cf. A027829, A033294.
%K A354256 nonn,base,more
%O A354256 1,1
%A A354256 _Jon E. Schoenfield_, May 21 2022
%E A354256 a(10) from _Chai Wah Wu_, May 24 2022