lambdaway :: rare

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_h1 rare numbers

_p Numbers when added or subtracted to its reverse gives a perfect square.

_ul [[https://rosettacode.org/wiki/Rare_numbers|https://rosettacode.org/wiki/Rare_numbers]]
_ul [[http://www.shyamsundergupta.com/rare.html|http://www.shyamsundergupta.com/rare.html]]
_ul [[https://www.planetmath.org/RareNumbers|https://www.planetmath.org/RareNumbers]]

{prewrap
Rare numbers are positive integers n where:

n   is expressed in base ten
  r   is the reverse of   n     (decimal digits)
  n   must be non-palindromic   (n ≠ r)
  (n+r) is the sum
  (n-r) is the difference and must be positive
  the sum and the difference must be perfect squares

Rare numbers exhibit certain properties:
- Rare numbers start always with an even digit and end in the digits 0, 2, 3, 7 and 8.
- Digital root of a rare number is always 2, 5, 8 and 9.

Consider a Rare number ABCD...MNPQ with any number of digits, then

1) Rare number can never start with an odd digit.

2) A-Q of any Rare number can only be 0, 1, 4, 5 or 6. 
   The value of Q can never be 1, 4, 6, 9

3) The digital root of a Rare number can take the values 2, 5, 8 or 9 only.
}

_h2 lambdatalk

{pre
'{def filter
 {lambda {:n}
  {let { {:a {W.first :n}}
         {:q {W.last :n}}
         {:d {- {W.first :n} {W.last :n}}}
       } {and {= {% :a 2} 0}
              {or {= :d 0} {= :d 1} {= :d 4} {= :d 5} {= :d 6}}
                  {not {or {= :q 1} {= :q 4} {= :q 6} {= :q 9}}}}}}}
-> {def filter
 {lambda {:n}
  {let { {:a {W.first :n}}
         {:q {W.last :n}}
         {:d {- {W.first :n} {W.last :n}}}
       } {and {= {% :a 2} 0}
              {or {= :d 0} {= :d 1} {= :d 4} {= :d 5} {= :d 6}}
                  {not {or {= :q 1} {= :q 4} {= :q 6} {= :q 9}}}}}}}

{filter 65}
{filter 621770}
{filter 281089082}
  {filter 281089083}
{filter 2022652202}
{filter 2042832002}

'{def lt_israre
 {lambda {:n}
  {let { {:n :n} 
         {:r {W.reverse :n}}
       } {if {and {> :n :r} 
                  {isInt {sqrt {+ :n :r}}}
                  {isInt {sqrt {- :n :r}}}}
          then :n
          else}}}}
-> {def lt_israre
 {lambda {:n}
  {if {filter :n}
   then {let { {:n :n} 
               {:r {W.reverse :n}}
             } {if {and {> :n :r} 
                        {isInt {sqrt {+ :n :r}}}
                        {isInt {sqrt {- :n :r}}}} then :n else}
             }
   else}}}

}

_p testing

{prewrap
{lt_israre 64}
{lt_israre 65}
{lt_israre 66}
{lt_israre 621770}

'{S.map lt_israre {S.serie 1 10000}}

'{S.map lt_israre {S.serie 1 700000}}
-> 65 621770              // computed in 7650ms

'{S.map lt_israre {S.serie 1 280000000}}
-> crushes my iPad...
}

_h2 calling javascript

{pre

LAMBDATALK.DICT["js_israres"] = function() '{
  var args = arguments[0].trim().split(" "),
      i0 = Number( args[0] ),
      i1 = Number( args[1] ),
      a = [];

var israre = function(n) {
    var r = Number( n.toString().split("").reverse().join("") );
    return (n > r) && (Number.isInteger(Math.sqrt(n+r))) 
                   && (Number.isInteger(Math.sqrt(n-r))) 
  };

for (var i=i0; i < i1; i++)
    if (israre(i)) a.push(i);
  return a   
};

}

_p testing

{prewrap
'{js_israres 1 2050000000} 
-> [65,621770,281089082,2022652202,2042832002]]  
 // computed in 784307ms ~ 13 minutes
}

_p {i alain marty | 2023/01/13}

{script 
LAMBDATALK.DICT["isInt"] = function() {
  var args = arguments[0];
  return (Number.isInteger(Number(args))) ? "true" : "false"
};

LAMBDATALK.DICT["js_israre"] = function() {
  var n = arguments[0],                      // "12345"
      r = n.split("").reverse().join(""),    // "54321"
      n = Number(n),
      r = Number(r);
  return (israre(n)) ? n : ""
};

var israre = function(n) {
  var r = Number( n.toString().split("").reverse().join("") );
  return (n > r) && (Number.isInteger(Math.sqrt(n+r))) 
                 && (Number.isInteger(Math.sqrt(n-r))) 
};

LAMBDATALK.DICT["js_israres"] = function() {
  var args = arguments[0].trim().split(" "),
      i0 = Number( args[0] ),
      i1 = Number( args[1] ),
      a = [];
  for (var i=i0; i < i1; i++)
    if (israre(i)) a.push(i);
  return a   
};

}