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

Author

Shyam Sunder Gupta, Dec 11 1999

Keywords

• nonn
• base
• nice

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

Examples

			65 - 56 = 9 and 65 + 56 = 121 are both squares.
		

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
• Shyam Sunder Gupta, Rare Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 14, 381-397. {{doi|10.1007/978-981-97-2465-9_14}}
• 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.

Crossrefs

Cf. A059755.

Programs

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 *)

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