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A035519
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Rare numbers: k-r and k+r are both perfect squares, where r is reverse of k and k is non-palindromic.
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5
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65, 621770, 281089082, 2022652202, 2042832002, 868591084757, 872546974178, 872568754178, 6979302951885, 20313693904202, 20313839704202, 20331657922202, 20331875722202, 20333875702202, 40313893704200
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OFFSET
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1,1
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COMMENTS
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There are 124 terms up to 10^22. - Shyam Sunder Gupta, Dec 15 2019
All non-palindromic numbers m = a^2 + b^2 such that reversal(m) = 2*a*b are terms of this sequence. For the numbers with this property, m - reversal(m) = (a-b)^2 and m + reversal(m) = (a+b)^2. - Metin Sariyar, Dec 19 2019
There are 132 terms up to 10^23. - Shyam Sunder Gupta, Nov 25 2021
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REFERENCES
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Shyam Sunder Gupta, Systematic computations of rare numbers, The Mathematics Education, Vol. XXXII, No. 3, Sept. 1998.
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LINKS
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Shyam Sunder Gupta, Table of n, a(n) for n = 1..132
Shyam Sunder Gupta, Rare Numbers
R. K. Guy, Conway's RATS and other reversals, Unsolved Problems Column, American Math. Monthly, page 425, May 1989.
R. K. Guy, Unsolved problems come of Age, American Math. Monthly, page 908, Dec. 1989.
Carlos Rivera, Conjecture 23. The Shyam's conjecture about the Rare Numbers, The Prime Puzzles and Problems Connection.
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EXAMPLE
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65 - 56 = 9 and 65 + 56 = 121 are both squares.
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MATHEMATICA
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r[n_]:=FromDigits[Reverse[IntegerDigits[n, 10]], 10]; f[n_]:=n!=r[n]&&IntegerQ[Sqrt[n-r[n]]]&&IntegerQ[Sqrt[n+r[n]]]; Timing[lst={}; Do[If[f[n], AppendTo[lst, n]], {n, 11, 15!}]; lst] (* Vladimir Joseph Stephan Orlovsky, Oct 10 2009 *)
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PROG
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(PARI) isok(k) = {my(d = digits(k), rd = Vecrev(d), r = fromdigits(rd)); (d != Vecrev(d)) && issquare(k-r) && issquare(k+r); } \\ Michel Marcus, Jan 06 2020
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CROSSREFS
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Cf. A059755.
Sequence in context: A015039 A238612 A185823 * A238842 A059755 A215657
Adjacent sequences: A035516 A035517 A035518 * A035520 A035521 A035522
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KEYWORD
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nonn,base,nice
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AUTHOR
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Shyam Sunder Gupta, Dec 11 1999
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STATUS
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approved
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