

A035519


Rare numbers: kr and k+r are both perfect squares, where r is reverse of k and k is nonpalindromic.


5



65, 621770, 281089082, 2022652202, 2042832002, 868591084757, 872546974178, 872568754178, 6979302951885, 20313693904202, 20313839704202, 20331657922202, 20331875722202, 20333875702202, 40313893704200
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OFFSET

1,1


COMMENTS

There are 124 terms up to 10^22.  Shyam Sunder Gupta, Dec 15 2019
All nonpalindromic 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) = (ab)^2 and m + reversal(m) = (a+b)^2. Some terms with this property: 65, 621770, 2042832002, 872546974178, 872568754178, 20313693904202, ...  Metin Sariyar, Dec 19 2019


REFERENCES

Shyam Sunder Gupta, Systematic computations of rare numbers, The Mathematics Education, Vol. XXXII, No. 3, Sept. 1998.


LINKS

Shyam Sunder Gupta, Table of n, a(n) for n = 1..124
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[nr[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 *)


PROG

(PARI) isok(k) = {my(d = digits(k), rd = Vecrev(d), r = fromdigits(rd)); (d != Vecrev(d)) && issquare(kr) && issquare(k+r); } \\ Michel Marcus, Jan 06 2020


CROSSREFS

Cf. A059755.
Sequence in context: A015039 A238612 A185823 * A238842 A059755 A215657
Adjacent sequences: A035516 A035517 A035518 * A035520 A035521 A035522


KEYWORD

nonn,base,nice


AUTHOR

Shyam Sunder Gupta, Dec 11 1999


STATUS

approved



