sequence with sum of digits

Number Theory Level 3

Let $s(n)$ denotes the sum of the digits of the positive integer $n$ . For example, $s(521)=5+2+1=8$ .

Define a sequence $\{a_n\}$ where $a_1=521$ and $a_{k+1}=a_k+s(a_k)$ for each $k\ge 1$ . Find the smallest value of $a_m$ for which $a_m>1000$ .

The answer is 1003.

2 solutions

Kyle T
Mar 12, 2019

<?php
$arr = array(521); //start with 521 as stated in question
do {
$a = $arr[count($arr)-1]; // get last value in array
$s = array_sum(str_split($a)); // sum of digits
$arr[] = $a+$s; //enter new value into sequence
} while($arr[count($arr)-1]<1000); //stop when we find a number > 1000
//print the answer
echo $arr[count($arr)-1].'<br>'; // 1003
//print the whole sequence
echo '<pre>'.print_r($arr,true).'</pre>'; // (see Joshua's answer for entire sequence)
?>

Joshua Lowrance
Mar 9, 2019

$\{521,529,545,559,578,598,620,628,644,658,677,697,719,738,752,766,785,805,818,835,851,865,884,904,917,934,950,964,983,1003\cdots \}$ . The first number over $1000$ is $1003$ .

sequence with sum of digits

The answer is 1003.

2 solutions

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