# Seven's Cycle Sum Sequence

Take a look at the sevens multiplication table from 7×0 to 7×9:

0, 7, 14, 21, 28, 35, 42, 49, 56, 63


If we just look at the digits in the one's place we get a permutation of the digits 0 through 9:

0, 7, 4, 1, 8, 5, 2, 9, 6, 3


Consider taking some positive decimal integer N and replacing each digit D in N with the the digit in the one's place of 7×D.

For example, 15209 becomes 75403 because 1 maps to 7, 5 maps to 5, 2 maps to 4, 0 maps to 0, and 9 maps to 3.

Now lets repeat this process with this new decimal integer until we see a cycle, i.e. until an integer we've already seen comes up.

For example, with 15209 we get the cycle

15209 -> 75403 -> 95801 -> 35607 -> 15209 -> repeats...
^
|
cycle restarts here


As another example, 505 has the short cycle

505 -> 505 -> repeats...
^
|
cycle restarts here


It turns out that for any N these cycles will always contain exactly 1 or 4 distinct integers. (I'll leave it to you to figure out why that is.) What's interesting is that if you sum all the distinct integer in a cycle, you almost always get a decimal integer that only consists of 2's and 0's.

For example, 15209 + 75403 + 95801 + 35607 = 222020.

N = 505 is one of the exceptions. The only integer in the cycle is 505 so the total sum is 505 itself.

Here are the sums of the cycles for N = 1 to 60:

N sum
1 20
2 20
3 20
4 20
5 5
6 20
7 20
8 20
9 20
10 200
11 220
12 220
13 220
14 220
15 220
16 220
17 220
18 220
19 220
20 200
21 220
22 220
23 220
24 220
25 220
26 220
27 220
28 220
29 220
30 200
31 220
32 220
33 220
34 220
35 220
36 220
37 220
38 220
39 220
40 200
41 220
42 220
43 220
44 220
45 220
46 220
47 220
48 220
49 220
50 50
51 220
52 220
53 220
54 220
55 55
56 220
57 220
58 220
59 220
60 200

We'll call this the Seven's Cycle Sum Sequence.

# Challenge

Write a program or function that takes in a positive decimal integer N and prints or returns, in decimal, the corresponding term of the Seven's Cycle Sum Sequence.

For example, if the input is 95801, the output should be 222020. If the input is 505, the output should be 505. If the input is 54, the output should be 220.

The shortest code in bytes wins.

• Of course, if you take the numbers from the one-cycles and multiply them by four, you'll find that all of them give numbers whose only digits are 2 and 0. Dec 7, 2015 at 9:30

## Python 2, 69 bytes

lambda n:[''.join('02'[x>'0']for x inn)+'0',n][set(n)<=set('05')]


The function is simple to describe:

• If n consists of only 0's and 5's, output it unchanged.
• Otherwise, replace each digit of n with 2, except 0 stays 0, and tack on a 0 to the end.

The golfing can be improved, I'm mostly posting to share the method. A language with native regex should allow a short solution.

An alternative statement of the function is

• In n, replace each digit with a 5, except 0 stays as 0
• If this changed n (it had a digit other than 0 or 5), multiply the result by 4

# Python 2, 63 bytes

lambda s:s.strip('05')and''.join((c>'0')*2for c in s)+'0'or s


Input argument is expected to be a string.

• Wow, I had no idea strip behaved this way.
– xsot
Dec 7, 2015 at 6:20
• Come on, type conversions (string &harr; number) are a substantial part of the fun (i.e. code length ;o)! Dec 8, 2015 at 10:48

## CJam, 16 bytes

Using the same algorithm as everyone else:

r_50s-{:~2fe&0}&


Test suite. (Generates all results from 1 to the input.)

### Explanation

r_      e# Read input and duplicate
50s     e# Push the string "50".
-       e# Remove all '5' and '0' characters from the input.
{       e# If any characters remained in the input...
:~    e#   Evaluate each digit character to turn it into an integer.
2fe&  e#   Map (&& 2) over the list. Due to short-circuiting, zeros remain zeros and
e#   everything else becomes 2.
0     e#   Push a trailing zero.
}&


# JavaScript (ES6), 54 51 bytes

Using xnor's method:

• 40 bytes: n=>n-(s=${n}.replace(/[^0]/g,5))?s*4:n Dec 8, 2015 at 9:55 • @charlie Wow, that s*4 trick is awesome! I reckon you should post this as a separate answer because the method is different enough and it's so much shorter than mine. :) Dec 8, 2015 at 10:02 • ok, i humbly will ;o) Dec 8, 2015 at 10:10 ## Mathematica, 8377 60 characters Tr@Union@NestList[FromDigits@Mod[7IntegerDigits@#,10]&,#,4]&  Ungolfed Tr@ Union@ NestList[ FromDigits@Mod[7 IntegerDigits@#, 10] &, #, 4 ] &  # JavaScript (ES5), 40 bytes n=>(s=${n}.replace(/[^0]/g,5))^n?s*4:n


It's an evolution of the user81655's solution, using the alternative approach described by xnor.

### Explanation

Sum of a non-zero digit in the 4-cycle is always 20, since the digit cycles either through 1→7→9→3, or 2→4→8→6, or 5→5→5→5. So replacing every such a digit with 5 doesn't change the sum.

That replacement action is reused to distinguish the 4-cycle from 1-cycle — if the replacement result is different from the input, then it's a 4-cycle, otherwise it's a 1-cycle.

N.B.: The template string ${n} is just for readability, (n+'') has the same length. • no regexp -- 47 bytes: n=>(s=[...${n}].map(d=>+d&&5).join)^n?s*4:n Dec 8, 2015 at 16:58

# Pyth, 14 bytes

s.uieM*R7jNTTQ


Not sure, why everybody determines the result by looking at patterns in the numbers. Simply doing the process, calculating all numbers of the circle and summing them up is shorter. At least in Pyth ;-)

Try it online: Demonstration or Test Suite

Btw, this is my 200th code-golf answer. So this post earns me the Gold code-golf badge.

### Explanation:

s.uieM*R7jNTTQ   implicit: Q = input number
.u          Q   apply the following expression to N=Q until it reaches a circle
jNT        convert N to base 10
*R7           multiply each digit with 7
eM              and perform modulo 10 for each number
i        T       convert digits from base 10 to a number
update N
.u returns the list of all intermediate results of N,
so we have now all numbers of the circle
s                sum them up

• Code... wait for it... golf! :) Congrats, and nice use of .u Dec 9, 2015 at 20:12

echo '501' | sed '/[^05]/{s/[^0]/2/g;s/$/0/}'2020 # Perl 6, 68 55 53 36 33 bytes {[+]$^a,{[~] $^b.comb.map: {'0741852963'.comb[$_]}}...^{$++*?/$a/}} # 68

{$_=@=$_.comb;[~] (@$_,(|.map(2*?+*),0))[$_⊈qw<0 5>]} # 55

{[~] ($_=@=$_.comb)⊆qw<0 5>??@$_!!(|.map(2*?+*),0)} # 53  {/^<[05]>+$/??$_!!S:g/./{2*?+$/}/~0} # 36


{m/^<[05]>+$/||S:g/./{2*?+$/}/~0} # 33


This is definitely the wrong way to do this, if the number is consists only of 5s and 0s it will return a Match object, otherwise it replaces everything but 0 with a 2, and append a 0 to the end.
( The Match object will behave like a number if you use it as one )

Though since it is doing it wrong, it makes it easy to point out the rare numbers by calling the gist method.

usage:

# give it a name
my &code = {...}

.say for (0..60,505,15209).flat.map({ code(\$_).gist.fmt: '%4s' }).rotor(1,10 xx 6,:partial)

( ｢0｣)
(  20   20   20   20  ｢5｣   20   20   20   20  200)
( 220  220  220  220  220  220  220  220  220  200)
( 220  220  220  220  220  220  220  220  220  200)
( 220  220  220  220  220  220  220  220  220  200)
( 220  220  220  220  220  220  220  220  220 ｢50｣)
( 220  220  220  220 ｢55｣  220  220  220  220  200)
(｢505｣)
(222020)


# Jelly, 9 bytes

D×7%⁵ƊƬḌS


Try it online!

## How it works

D×7%⁵ƊƬḌS - Main link. Takes an integer n on the left
D         - Convert to digits
ƊƬ   - Repeat the following until a value is repeated and collect all intermediate values:
×7       -   Multiply each digit by 7
%⁵     -   Take the ones digit
Ḍ  - Convert each list of digits back to a number
S - Take the sum