cp's OEIS Frontend

A068536 Numbers m such that m^2 + (reversal of m)^2 is a square. (Leading 0's are ignored.)

%I A068536 #33 May 02 2023 09:00:59
%S A068536 88209,90288,125928,196020,368280,829521,1978020,2328480,5513508,
%T A068536 8053155,19798020,86531940,197998020,554344560,556326540,1960396020,
%U A068536 1979998020,5543944560,5925169800,8820988209,9028890288,12592925928,14011538112,19602196020,19799998020
%N A068536 Numbers m such that m^2 + (reversal of m)^2 is a square. (Leading 0's are ignored.)
%C A068536 The sequence is infinite, even if it is restricted to terms that end with a nonzero digit, as any term generates an infinite number of other terms by the following scheme: If m is a term of the sequence and d(m) denotes the number of digits of m, then set m' = m*10^d'+m with d' >= d(m). For d' >= d(m) we have reverse(m') = reverse(m)*10^d' + reverse(m) and thus (m')^2 + reverse(m')^2 = (m*10^d'+m)^2 + (reverse(m)*10^d'+reverse(m))^2 = (m^2+reverse(m)^2)*(10^d'+1)^2. As m^2+reverse(m)^2 is a perfect square by assumption, the product on the right-hand side of the equation is also a perfect square and m' is part of the sequence. The calculation works also with m' = m*(10^(k*d')+...+10^d'+1). As an example take a(1)=88209. All numbers 8820988209, 882098820988209, 88209882098820988209, ... and 88209088209, 882090088209, 8820900088209, ... are also terms of the sequence. - _Matthias Baur_, May 01 2020
%H A068536 Giovanni Resta, <a href="/A068536/b068536.txt">Table of n, a(n) for n = 1..60</a>
%H A068536 Sheng Jiang and Rui-Chen Chen, <a href="https://doi.org/10.1080/0020739980290505">Digits reversed Pythagorean triples</a>, International Journal of Mathematical Education in Science and Technology, volume 29, number 5, 1998, pages 689-696.
%e A068536 88209^2 + 90288^2 = 126225^2, so 88209 belongs to the sequence.
%t A068536 Do[If[IntegerQ[Sqrt[n^2 + FromDigits[Reverse[IntegerDigits[n]]]^2]], Print[n]], {n, 1, 10^6}]
%Y A068536 Cf. A004086 (digit reversal), A202386, A256515.
%K A068536 base,nonn
%O A068536 1,1
%A A068536 _Joseph L. Pe_, Mar 22 2002
%E A068536 a(7)-a(15) from _Donovan Johnson_, Apr 09 2010
%E A068536 a(16)-a(25) from _Donovan Johnson_, Jul 15 2011