A256515 -id:A256515 - cp's OEIS frontend

A202386 Nonpalindromic numbers m such that the difference between the square of m and the square of the reversal of m is itself a perfect square. Numbers ending in 0 are excluded.

65, 5625, 6565, 50721, 65065, 71555, 75515, 84295, 541063, 557931, 650065, 650606, 656565, 699796, 809325, 827372, 934065, 2855182, 4637061, 4854634, 5791775, 5883141, 5951693, 6129084, 6500065, 6731076, 6752626, 6791774, 7768827, 8084505, 9349065

Offset: 1

Arkadiusz Wesolowski, Dec 18 2011

This sequence is infinite because 65*10^k + 65 is a term for all k > 1.

			5625 belongs to this sequence because 5625^2 - 5265^2 = 1980^2.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1996, p. 147.

Giovanni Resta, Table of n, a(n) for n = 1..200
Sheng Jiang and Rui-Chen Chen, Digits reversed Pythagorean triples, International Journal of Mathematical Education in Science and Technology, volume 29, number 5, 1998, pages 689-696, see type acca-DRPT.

Cf. A000290 (squares), A004086 (digit reversal).

Cf. A256515 (with abs), A068536 (with addition).

lst = {}; Do[a = n^2; b = FromDigits[Reverse[IntegerDigits[n]]]^2; If[MatchQ[Sqrt[a - b], _Integer] && ! a == b, AppendTo[lst, n]], {n, 85000}]; Select[lst, ! Mod[#, 10] == 0 &]

isok(m) = my(r=fromdigits(Vecrev(digits(m)))); (r != m) && (m % 10) && issquare(m^2 - r^2); \\ Michel Marcus, Feb 27 2020

Name clarified by Michel Marcus, Feb 27 2020

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

Original entry on oeis.org

88209, 90288, 125928, 196020, 368280, 829521, 1978020, 2328480, 5513508, 8053155, 19798020, 86531940, 197998020, 554344560, 556326540, 1960396020, 1979998020, 5543944560, 5925169800, 8820988209, 9028890288, 12592925928, 14011538112, 19602196020, 19799998020

Offset: 1

Joseph L. Pe, Mar 22 2002

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

			88209^2 + 90288^2 = 126225^2, so 88209 belongs to the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..60
Sheng Jiang and Rui-Chen Chen, Digits reversed Pythagorean triples, International Journal of Mathematical Education in Science and Technology, volume 29, number 5, 1998, pages 689-696.

Cf. A004086 (digit reversal), A202386, A256515.

Do[If[IntegerQ[Sqrt[n^2 + FromDigits[Reverse[IntegerDigits[n]]]^2]], Print[n]], {n, 1, 10^6}]

a(7)-a(15) from Donovan Johnson, Apr 09 2010

a(16)-a(25) from Donovan Johnson, Jul 15 2011

A289140 Positive numbers k such that rev(k)^2 + rev(k^2) is a square, where rev(n) = A004086(n) is the digital reverse of n.

Original entry on oeis.org

998586, 3632658, 9985860, 36326580, 74471091, 99664458, 99858600, 363265800, 634826115, 743193501, 744710910, 756335085, 759317343, 996644580, 998586000, 3632658000, 6348261150, 7177621788, 7431935010, 7447109100, 7563350850, 7593173430, 9966445800

Offset: 1

Giovanni Resta, Jun 26 2017

Every term must be a multiple of 3.

			998586 is a term since rev(998586^2) + 685899^2 = 1079100^2.

Cf. A004086, A061230, A061231, A068536, A102859, A113798, A202386, A256515.

rev[n_] := FromDigits@ Reverse@ IntegerDigits@ n; Parallelize@ Select[3 Range[4 10^6], IntegerQ@ Sqrt[rev[#^2] + rev[#]^2] &]

isok(n) = issquare(fromdigits(Vecrev(digits(n)))^2 + fromdigits(Vecrev(digits(n^2)))); \\ Michel Marcus, Jun 29 2017

cp's OEIS Frontend

A202386 Nonpalindromic numbers m such that the difference between the square of m and the square of the reversal of m is itself a perfect square. Numbers ending in 0 are excluded.

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

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A289140 Positive numbers k such that rev(k)^2 + rev(k^2) is a square, where rev(n) = A004086(n) is the digital reverse of n.

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