Two palindromes are not enough

Question

Some numbers, such as \$14241\$, are palindromes in base 10: if you write the digits in reverse order, you get the same number.

Some numbers are the sum of 2 palindromes; for example, \$110=88+22\$, or \$2380=939+1441\$.

For other numbers, 2 palindromes are not enough; for example, 21 cannot be written as the sum of 2 palindromes, and the best you can do is 3: \$21=11+9+1\$.

Write a function or program which takes integer input n and outputs the nth number which cannot be decomposed as the sum of 2 palindromes. This corresponds to OEIS A035137.

Single digits (including 0) are palindromes.

Standard rules for sequences apply:

• input/output is flexible
• you may use 0- or 1- indexing
• you may output the nth term, or the first n terms, or an infinite sequence

(As a sidenote: all integers can be decomposed as the sum of at most 3 palindromes.)

Test cases (1-indexed):

1 -> 21
2 -> 32
10 -> 1031
16 -> 1061
40 -> 1103

This is code-golf, so the shortest answer wins.

Answer 1

JavaScript (ES6),  93 83 80  79 bytes

Saved 1 byte thanks to @tsh

Returns the \$n\$^th term, 1-indexed.

i=>eval("for(n=k=1;k=(a=[...k+[n-k]+k])+''!=a.reverse()?k-1||--i&&++n:++n;);n")

Try it online!

How?

Given \$n\$, we test whether there exists any \$1\le k\le n\$ such that both \$k\$ and \$n-k\$ are palindromes. If we do find such a \$k\$, then \$n\$ is the sum of two palindromes.

The trick here is to process \$k\$ and \$n-k\$ at the same time by testing a single string made of the concatenation of \$k\$, \$n-k\$ and \$k\$.

Example:

For \$n=2380\$:

• we eventually reach \$k=1441\$ and \$n-k=939\$
• we test the string "\$1441\color{red}{939}1441\$" and find out that it is a palindrome

Commented

NB: This is a version without eval() for readability.

i => {                       // i = index of requested term (1-based)
  for(                       // for loop:
    n = k = 1;               //   start with n = k = 1
    k =                      //   update k:
      ( a =                  //     split and save in a[] ...
        [...k + [n - k] + k] //     ... the concatenation of k, n-k and k
      ) + ''                 //     coerce it back to a string
      != a.reverse() ?       //     if it's different from a[] reversed:
        k - 1                //       decrement k; if the result is zero:
          || --i             //         decrement i; if the result is not zero:
            && ++n           //           increment n (and update k to n)
                             //         (otherwise, exit the for loop)
      :                      //     else:
        ++n;                 //       increment n (and update k to n)
  );                         // end of for
  return n                   // n is the requested term; return it
}                            //

Answer 2

Jelly,  16 10  9 bytes

-1 byte thanks to Erik the Outgolfer. Outputs the first \$n\$ terms.

2_ŒḂƇ⁺ṆƲ#

Try it online!

I tried to come up with different idea compared to my original approach. Let's review the thinking process:

• Initially, the test worked as follows: It generated the integer partitions of that number, then filtered out those that also contained non-palindromes, then counted how many length-2 eligible lists there were. This was obviously not too efficient in terms of code length.

• Generating the integer partitions of \$N\$ and then filtering had 2 main disadvantages: length and time efficiency. To solve that issue, I thought I shall first come up with a method to generate only the pairs of integers \$(x, y)\$ that sum to \$N\$ (not all arbitrary-length lists) with the condition that both numbers must be palindrome.

• But still, I wasn't satisfied with the "classic way" of going about this. I switched approaches: instead of generating pairs, let's have the program focus on idividual palindromes. This way, one can simply compute all the palindromes \$x\$ below \$N\$, and if \$N-x\$ is also palindrome, then we're done.

Code Explanation

2_ŒḂƇ⁺ṆƲ# – Monadic link or Full program. Argument: n.
2       # – Starting at 2*, find the first n integers that satisfy...
 _ŒḂƇ⁺ṆƲ  – ... the helper link. Breakdown (call the current integer N):
    Ƈ     – Filter. Creates the range [1 ... N] and only keeps those that...
  ŒḂ      – ... are palindromes. Example: 21 -> [1,2,3,4,5,6,7,8,9,11]
 _        – Subtract each of those palindromes from N. Example: 21 -> [20,19,...,12,10]
     ⁺    – Duplicate the previous link (think of it as if there were an additional ŒḂƇ
            instead of ⁺). This only keeps the palindromes in this list.
            If the list is non-empty, then that means we've found a pair (x, N-x) that
            contains two palindromes (and obviously x+N-x=N so they sum to N).
      Ṇ   – Logical NOT (we're looking for the integers for which this list is empty).
       Ʋ  – Group the last 4 links (basically make _ŒḂƇ⁺Ṇ act as a single monad).

^{* Any other non-zero digit works, for that matter.}

Answer 3

Jelly, 11 bytes

⁵ŻŒḂ€aṚ$EƲ#

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The full program roughly works like this:

1. Set z to the input.
2. Set x to 10.
3. Set R to [].
4. For every integer k from 0 up to and including x, check whether both k and x - k are palindromic.
5. If all elements of L are equal (that is, if either all possible pairs that sum to x have both their elements palindromic, or all such pairs have at most one of their elements be palindromic), set z to z - 1 and append x to R.
6. If z = 0, return R and end.
7. Set x to x + 1.
8. Go to step 4.

You may suspect that step 5 doesn't actually do the job it should. We should really not decrement z if all pairs that sum to x are palindromic. However, we can prove that this will never happen:

Let's first pick an integer \$k\$ so that \$10\le k\le x\$. We can always do so, because, at step 2, we initialize x to be 10.

If \$k\$ isn't a palindrome, then we have the pair \$(k,x-k)\$, where \$k+(x-k)=x\$, therefore not all pairs have two palindromes.

If, on the other hand, \$k\$ is a palindrome, then we can prove that \$k-1\$ isn't a palindrome. Let the first and last digits of \$k\$ be \$D_F\$ and \$D_L\$ respectively. Since \$k\$ is a palindrome, \$D_F=D_L>0\$. Let the first and last digits of \$k-1\$ be \$D'_F\$ and \$D'_L\$ respectively. Since \$D_L>0\$, \$D'_L=D'_F-1\ne D'_F\$. Therefore, \$k-1\$ isn't a palindrome, and we have the pair \$(k-1,x-(k-1))\$, where \$(k-1)+(x-(k-1))=k-1+x-k+1=x\$.

We conclude that, if we start with setting x to a value greater than or equal to 10, we can never have all pairs of non-negative integers that sum to x be pairs of palindromes.

Answer 4

Retina, 135 102 bytes

K`0
"$+"{0L$`\d+
*__
L$`
<$.'>$.`>
/<((.)*.?(?<-2>\2)*(?(2)$)>){2}/{0L$`\d+
*__
))L$`
<$.'>$.`>
0L`\d+

Try it online! Too slow for n of 10 or more. Explanation:

K`0

Start off by trying 0.

"$+"{

Repeat n times.

0L$`\d+
*__

Convert the current trial value to unary and increment it.

L$`
<$.'>$.`>

Create all pairs of non-negative integers that sum to the new trial value.

/<((.)*.?(?<-2>\2)*(?(2)$)>){2}/{

Repeat while there exists at least one pair containing two palindromic integers.

0L$`\d+
*__
))L$`
<$.'>$.`>

Increment and expand the trial value again.

0L`\d+

Extract the final value.

Answer 5

Haskell, 68 67 63 bytes

[n|n<-[1..],and[p a||p(n-a)|a<-[0..n]]]
p=((/=)=<<reverse).show

Returns an infinite sequence.

Collect all n where either a or n-a is not a palindrome for all a <- [0..n].

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Answer 6

Perl 5 -MList::Util=any -p, 59 55 bytes

-3 bytes thanks to @NahuelFouilleul

++$\while(any{$\-reverse($\-$_)==reverse}0..$\)||--$_}{

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Note: any could be replaced by grep and avoid the -M command line switch, but under the current scoring rules, that would cost one more byte.

Answer 7

R, 115 111 bytes

-4 thanks to Giuseppe

function(n,r=0:(n*1e3))r[!r%in%outer(p<-r[Map(Reduce,c(x<-paste0),Map(rev,strsplit(a<-x(r),"")))==a],p,'+')][n]

Try it online!

Most of the work is packed into the function arguments to remove the {} for a multi-statement function call, and to reduce the brackets needed in defining the object r

Basic strategy is to find all palindromes up to a given bound (including 0), find all pairwise sums, and then take the n-th number not in that output.

The bound of n*1000 was chosen purely from an educated guess, so I encourage anyone proving/disproving it as a valid choice.

r=0:(n*1e3)can probably be improved with a more efficient bound.

Map(paste,Map(rev,strsplit(a,"")),collapse="")is ripped from Mark's answer here, and is just incredibly clever to me.

r[!r%in%outer(p,p,'+')][n]reads a little inefficient to me.

Answer 8

C# (Visual C# Interactive Compiler), 124 bytes

n=>{int a=0;for(string m;n>0;)if(Enumerable.Range(0,++a).All(x=>!(m=x+""+(a-x)+x).Reverse().SequenceEqual(m)))n--;return a;}

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Answer 9

J, 57/60 bytes

0(](>:^:(1&e.p e.]-p=:(#~(-:|.)&":&>)&i.&>:)^:_)&>:)^:[~]

Try it online!

The linked version adds 3 bytes for a total of 60 in order to save as a function that the footer can call.

In the REPL, this is avoided by calling directly:

   0(](>:^:(1 e.q e.]-q=:(#~(-:|.)&":&>)&i.&>:)^:_)&>:)^:[~] 1 2 10 16 40
21 32 1031 1061 1103

Explanation

The general structure is that of this technique from an answer by Miles:

(s(]f)^:[~]) n
          ]  Gets n
 s           The first value in the sequence
         ~   Commute the argument order, n is LHS and s is RHS
        [    Gets n
      ^:     Nest n times with an initial argument s
  (]f)         Compute f s
         Returns (f^n) s

This saved a few bytes over my original looping technique, but since the core function is my first attempt at writing J, there is likely still a lot that can be improved.

0(](>:^:(1&e.p e.]-p=:(#~(-:|.)&":&>)&i.&>:)^:_)&>:)^:[~]
0(]                                                 ^:[~] NB. Zero as the first term switches to one-indexing and saves a byte.
   (>:^:(1&e.p e.]-p=:(#~(-:|.)&":&>)&i.&>:)^:_)&>:)      NB. Monolithic step function.
                                                 >:       NB. Increment to skip current value.
   (>:^: <predicate>                        ^:_)          NB. Increment current value as long as predicate holds.
                   p=:(#~(-:|.)&":&>)&i.&>:               NB. Reused: get palindromes in range [0,current value].
                       #~(-:|.)&":&>                      NB. Coerce to strings keeping those that match their reverse.
                 ]-p                                      NB. Subtract all palindromes in range [0,current value] from current value.
    >:^:(1&e.p e.]-p                                      NB. Increment if at least one of these differences is itself a palindrome.

Answer 10

05AB1E, 15 12 bytes

°ÝDʒÂQ}ãOKIè

-3 bytes thanks to @Grimy.

0-indexed.
Very slow, so times out for most test cases.

Try it online or verify the first few cases by removing the Iè.

Much faster previous 15 byter version:

µNÐLʒÂQ}-ʒÂQ}g_

1-indexed.

Try it online or output the first \$n\$ values.

Explanation:

°Ý              # Create a list in the range [0, 10**input]
  D             # Duplicate this list
   ʒÂQ}         # Filter it to only keep palindromes
       ã        # Take the cartesian product with itself to create all possible pairs
        O       # Sum each pair
         K      # Remove all of these sums from the list we duplicated
          Iè    # Index the input-integer into it
                # (after which the result is output implicitly)

µ               # Loop until the counter variable is equal to the (implicit) input-integer
 NÐ             #  Push the loop-index three times
   L            #  Create a list in the range [1, N] with the last copy
    ʒÂQ}        #  Filter it to only keep palindromes
        -       #  Subtract each from N
         ʒÂQ}   #  Filter it again by palindromes
             g_ #  Check if the list is empty
                #   (and if it's truthy: increase the counter variable by 1 implicitly)
                # (after the loop: output the loop-index we triplicated implicitly as result)

Answer 11

Red, 142 bytes

func[n][i: 1 until[i: i + 1 r: on repeat k i[if all[(to""k)= reverse
to""k(s: to""i - k)= reverse copy s][r: off break]]if r[n: n - 1]n < 1]i]

Try it online!

Returns n-th term, 1-indexed

Answer 12

Python 3, 107 bytes

p=lambda n:str(n)!=str(n)[::-1]
def f(n):
 m=1
 while n:m+=1;n-=all(p(k)+p(m-k)for k in range(m))
 return m

Try it online!

Inverting the palindrome checking saved 2 bytes :)

For reference the straight forward positive check (109 bytes):

p=lambda n:str(n)==str(n)[::-1]
def f(n):
 m=1
 while n:m+=1;n-=1-any(p(k)*p(m-k)for k in range(m))
 return m

Answer 13

APL(NARS), 486 bytes

r←f w;p;i;c;P;m;j
p←{k≡⌽k←⍕⍵}⋄i←c←0⋄P←r←⍬
:while c<w
    i+←1
    :if   p i⋄P←P,i⋄:continue⋄:endif
    m←≢P⋄j←1
    :while j≤m
         :if 1=p i-j⊃P⋄:leave⋄:endif
         j+←1
    :endwhile
    :if j=m+1⋄c+←1⋄r←i⋄:endif
:endwhile

What is the word for break the loop? It seems it is ":leave", right? {k≡⌽k←⍕⍵} in p is the test for palindrome. This above function in the loop store all the palindrome found in the set P, if for some element w of P is such that i-w is in P too this means that the i is not right and we have increment i. Results:

  f 1
21
  f 2
32
  f 10
1031
  f 16
1061
  f 40
1103
  f 1000
4966
  f 1500
7536