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A035519 Rare numbers: k-r and k+r are both perfect squares, where r is reverse of k and k is non-palindromic. 5
65, 621770, 281089082, 2022652202, 2042832002, 868591084757, 872546974178, 872568754178, 6979302951885, 20313693904202, 20313839704202, 20331657922202, 20331875722202, 20333875702202, 40313893704200 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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
REFERENCES
Shyam Sunder Gupta, Systematic computations of rare numbers, The Mathematics Education, Vol. XXXII, No. 3, Sept. 1998.
LINKS
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.
EXAMPLE
65 - 56 = 9 and 65 + 56 = 121 are both squares.
MATHEMATICA
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 *)
Select[Range[2043*10^6], !PalindromeQ[#]&&AllTrue[{Sqrt[#+ IntegerReverse[ #]], Sqrt[ #-IntegerReverse[#]]}, IntegerQ]&] (* The program generates the first 5 terms of the sequence. *) (* Harvey P. Dale, Jan 22 2023 *)
PROG
(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
CROSSREFS
Cf. A059755.
Sequence in context: A015039 A238612 A185823 * A368189 A238842 A059755
KEYWORD
nonn,base,nice
AUTHOR
Shyam Sunder Gupta, Dec 11 1999
STATUS
approved

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Last modified April 23 18:16 EDT 2024. Contains 371916 sequences. (Running on oeis4.)