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

1,1

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

Links

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

Example

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.

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 *)

Prog

(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)

Crossrefs

Cf. A000404, A004086, A061205.

Sequence in context: A331062 A313666 A242128 * A317045 A037977 A044845

Adjacent sequences: A270923 A270924 A270925 * A270927 A270928 A270929

Keyword

base,nonn,easy

Author

Soumil Mandal, Mar 26 2016

Extensions

More terms from Altug Alkan, Mar 26 2016

Status

approved