A270926 Numbers k such that k*R(k) can be represented as the sum of two nonzero squares, where R(k) is the reverse of the decimal expansion of k.

5, 10, 15, 16, 18, 20, 25, 30, 37, 40, 50, 51, 52, 55, 58, 60, 61, 70, 73, 78, 80, 81, 85, 87, 89, 90, 98, 100, 101, 104, 106, 109, 110, 111, 122, 125, 128, 145, 146, 148, 149, 150, 159, 160, 162, 164, 165, 168, 169, 174, 176, 180, 181, 192, 195, 198, 200, 202, 208, 212, 220, 221, 222

Offset: 1

Author

Soumil Mandal, Mar 26 2016

Keywords

• base
• nonn
• easy

Comments

k*R(k) is the square of the hypotenuse of a right triangle.

Palindromes in this sequence are 5, 55, 101, 111, 181, 202, 212, 222, 232, 272, 292, 303, 313, 323, 333, 353, 373, ... - Altug Alkan, Mar 26 2016

Examples

			For k=5, R(k)=5 and k*R(k)=25, which is 3^2 + 4^2.
For k=10, R(k)=1 and k*R(k)=10, which is 1^2 + 3^2.
For k=58, R(k)=85 and k*R(k)=4930, which is 13^2 + 69^2.
		

Links

• Paolo P. Lava, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000404, A004086, A061205.

Programs

Mathematica

Select[Range@ 222, Length[PowersRepresentations[# FromDigits@ Reverse@ IntegerDigits@ #, 2, 2] /. {0, } -> Nothing] > 0 &] (* _Michael De Vlieger, Mar 26 2016 *)
stnzsQ[{a_,b_}]:=AllTrue[{a,b},IntegerQ[Sqrt[#]]&]; Select[Range[ 250], Length[ Select[IntegerPartitions[#  IntegerReverse[#],{2}],stnzsQ]] >0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 12 2020 *)

PARI

isA000404(n) = {for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2))}
lista(nn) = for(n=1, nn, if(isA000404(n*eval(concat(Vecrev(Str(n))))), print1(n, ", "))); \\ Altug Alkan, Mar 26 2016

Python

# Soumil Mandal, Mar 27 2016
def isHypotenuse(num):
    a, b = 1, 1
    a2, b2 = a**2, b**2
    while a2 + b2 <= num:
        while a2 + b2 <= num:
            if a2 + b2 == num:
                return True
            b += 1
            b2 = b**2
        a += 1
        a2 = a**2
        b = 1
        b2 = b**2
    return False
for x in range(20000):
    if isHypotenuse(x * int(str(x)[::-1])):
        print(x)

Extensions

More terms from Altug Alkan, Mar 26 2016