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Let's take the number 180. This is an interesting number because the sum of digits of this number is equal to:

1 + 8 + 0 = 9

And the squared version of this number, or:

180² = 32400 > 3 + 2 + 4 + 0 + 0 = 9

These are both 9. The sum of digits of the original number and the squared number are the same. Of course, this is also found at OEIS: A058369.


Given a non-negative integer n, output the nth positive number with this condition.

Test cases (zero-indexed)

Input > Output

0 > 1
1 > 9
2 > 10
3 > 18
4 > 19
5 > 45
6 > 46
7 > 55
8 > 90
9 > 99
10 > 100
11 > 145
12 > 180
13 > 189
14 > 190
15 > 198
16 > 199
17 > 289
18 > 351
19 > 361

The input can also be 1-indexed if that fits you better.

This is , so the submission with the least amount of bytes wins!

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In case nobody has spotted it yet, only numbers which are equivalent to 0 or 1 (mod 9) can appear in the list. – Neil Mar 12 at 19:05
@MamaFunRoll Um... no. Sorry. Numbers with digital roots of 5 have squares whose digital root is 7. – Neil Mar 12 at 20:17
@Neil owait nvm – Mama Fun Roll Mar 12 at 20:42

12 Answers 12

up vote 5 down vote accepted

Jelly, 13 bytes


Input is 1-indexed. Try it online!

How it works

1dz#Ṫ    Main link. Argument: n (index)

1        Set the return value to 1.
   #     Execute ... until ... matches have been found.
 Ç         the helper link
  ³        n
    Ṫ    Extract the last match.

,²DS€=/  Helper link. Argument: k (integer)

,²       Pair k with k².
  D      Convert each to decimal.
   S€    Compute the sum of each list of base 10 digits.
     =/  Reduce by equality.
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Haskell, 54 bytes
([x|x<-[1..],s x==s(x^2)]!!)

Usage example: ([x|x<-[1..],s x==s(x^2)]!!) 17 -> 289.

s calculates the digit sum:

                    show     -- turn number into a string
     map(read.pure)          -- turn every character (the digits) in to a
                             -- one element string and convert back to integer
sum                          -- sum those integers

main function:

[x|x<-[1..]            ]     -- make a list of all x starting from 1
           ,s x==s(x^2)      -- where s x == s (x^2)
                        !!   -- pick nth element from that list
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Perl 6, 47 46 bytes

{(grep {$_.comb.sum==$_².comb.sum},1..*)[$_]}
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Mathematica, 64 bytes


Simple anonymous function. Zero-indexed.

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JavaScript (ES6), 76 73 72 bytes


I spent 30 minutes trying to get this to work until I realized I was outputting the wrong variable :|

This is zero-indexed.

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I feel like turning this into a recursive function would shorten this up a lot... – Mama Fun Roll Mar 12 at 3:00

Pyth, 15


1 byte thanks to DenkerAffe!

Try it here or run a Test Suite.

Uses the 1-indexed option.

Naive implementation using .f which gets the first n numbers that match the given condition.

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You can save one byte by removing h if you use 1-indexing which is explicitly allowed. – DenkerAffe Mar 11 at 22:29
@DenkerAffe Oh, thanks I should read more closely :P – FryAmTheEggman Mar 11 at 22:33

MATL, 24 23 bytes


Uses 1-based input.

Try it online!

x        % take inpout and delete it (gets copied into clipboard G)
`        %   do...while
  @      %   push loop iteration index: candidate number, n
  2:^    %   array [n n^2]
  "      %   for each element of that array 
    @    %     push that element 
    V!U  %     get its digits (to string, transpose, to number)
    Xs   %     compute their sum
  ]      %   end for each
  =      %   are the two sums equal?
  ?      %   if so
    @    %     the candidate number is valid: push it
  ]      %   end if
  NG<    %   is number of elements in stack less than input?
]        % if so, proceed with next iteration. End do...while. 
1$       % specify 1 input for implicit display: only top of stack
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very nice that MATL is finally listed among distant compilers there !. – delete me here Mar 12 at 12:39
@Agawa001 Thanks! :-) – Luis Mendo Mar 12 at 14:00

Julia, 79 66 bytes


This is a recursive function that accepts an integer and returns an integer. It uses 1-based indexing.

We store a few things as function arguments:

  • n : The input
  • x : A counter for how many numbers with this condition we've found
  • i : A number to check for the condition
  • s : A function to compute the sum of the digits of its input

While x is less than the input, we recurse, incrementing x if i meets the condition and incrementing i. Once x == n, we return i, but we have to subtract 1 because it will have been incremented one too many times.

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Convex 0.2, 36 35 bytes

Convex is a new language that I am developing that is heavily based on CJam and Golfscript. The interpreter and IDE can be found here. Input is an integer into the command line arguments. Indexes are one-based. Uses the CP-1252 encoding.

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Mathematica, 63 60 61 59 bytes


While making this the other answer popped up but I'm beating them by a single byte and I'm posting this before that one gets golfed. One indexed.

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Fails for input >2457. Simply increasing your Range won't help, because A058369[n]/n doesn't seem to converge. – murphy Mar 11 at 23:14
Better? filler+ – CalculatorFeline Mar 11 at 23:31
10^# would be shorter than 2^#*9. Of course it becomes too slow after n is bigger than about 6... – feersum Mar 12 at 3:19
Why not 9^#?fil – CalculatorFeline Mar 12 at 3:30
Do you have a proof that f(n) <= 9^n? (10 is obvious because 10^n is always a solution). – feersum Mar 12 at 3:42

Retina, 103 bytes

$*1 x
}`^ ?x


Definitely golfable.

Uses the new Retina feature % for squaring (hence not working with the online version yet).

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Mathcad, 70 50 bytes

Mathcad has no built in functions to convert a number to its digit string, so the user function d(a) does this job. A program then iterates through the positive integers, testing for equality of sums, until it has accumulated n numbers in the vector v. The program is evaluated using the = operator, which displays the result vector. (Note that the whole program appears exactly as displayed below on the Mathcad worksheet)

Updated program: Assumes default initialization of a to zero and makes use of fact that Mathcad returns the value of the last evaluated statement in a program.
Makes use of evaluation order of expressions to increment variable a in the first summation (and which is then available for use in the sum of square)

enter image description here

Original program: Returns a vector of all numbers up to n.

enter image description here

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