A035519 Rare numbers: k-r and k+r are both perfect squares, where r is reverse of k and k is non-palindromic.

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

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, 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.

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

Adjacent sequences: A035516 A035517 A035518 * A035520 A035521 A035522

Keyword

nonn,base,nice

Author

Shyam Sunder Gupta, Dec 11 1999

Status

approved