Partitioning reciprocals

Question

Given a number n > 77, write a program or function that finds a set of distinct positive integers such that the sum of the set equals n, and the sum of the reciprocals of the set equals 1.

Example for 80:

80 = 2 + 4 + 10 + 15 + 21 + 28    ⟶     1/2 + 1/4 + 1/10 + 1/15 + 1/21 + 1/28 = 1

Your program or function must (theoretically) work for any n < 2³², and is not excused for floating point rounding errors. Note that solutions exist for all n > 77.

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Shortest code in bytes wins.

There is a bonus incentive: I will award a bounty to the smallest solution that works for any n and runs log(n). For small n it must be fast (determined at my discretion). Yes, this is possible.

Answer 1

Mathematica, 54 bytes

Select[IntegerPartitions@#,Unequal@@#&&Tr[1/#]==1&,1]&

About as inefficient as it gets, but it does solve n = 78 in about 9 seconds.

The result is returned as a list wrapped in a singleton list, e.g.:

{{45, 12, 9, 5, 4, 3}}

Answer 2

Python 3, 7306 1995 Bytes

This solution runs in log(n) complexity (as far as I can tell).

def i(s,t):
 for n in s[::-1]:t=t.replace(*n)
 return [[]]*78+[list(bytearray.fromhex(a))for a in t.split(",")]
def f(n):
 g,h=lambda c,n:c+[[[2],[3,7,78,91]][n[len(c)]%2]+[i*2for i in c[-1]]],lambda n:[]if n<78 else h((n-[2,179][n%2])//2)+[n]
 v=h(n);c=[i([['g',',03040'],['h',',,0306080'],['i',',020'],['j','b0c1'],['k','21'],['l','60'],['m','30'],['n','141'],['o','k24'],['p',',g'],['q','618'],['r','0c0'],['s','1e'],['t',',0ml'],['u','283c'],['v','e0f1'],['w','2a38'],['x','80'],['y','a0'],['z','01'],['A','50'],['B','24'],['C','i40'],['D','plb1'],['E','gl'],['F','48'],['G','bre1'],['H','28'],['I','6k'],['J','416s'],['K',',040Al'],['L','90'],['M','2a'],['N','54'],['O','k6o'],['P','3c'],['Q','il'],['R','18'],['S','px'],['T','im'],['U','70'],['V','b1'],['W','23'],['X','pj'],['Y','hj'],['Z','0n']],'020lxycHTaRHCyf1517CyfneC91k51cCLdneQU912MCyf0dBiALyf2dClfPEyfneT9s2dELdneEjIgmLydHg5rd14BKLardsE3n8sQ9rd1517Q9rdneplmdRBgUmcRMC5sPEyf102bgA6sPE91z2miAj41IQmc0dRBQUen7spl31z82bT9RFT3wE7neMgmyf0dRBgUmaHMELc1b36EUdBMQLyfs2d,C710M2bgLardRHT3BFQ9rf0dPQ7rdBMQm9Rs2d,0mAl9100d142bE710M2bQmc0fRPtxarfn8sEc1k4sBTfnePExcwtxarf1k8BExcuT3kkT91663C51964,0mAl71k4BMELe12NTcRwQjOT820ltmarf1z8mExeRNCqBFtmyjIHKLa100ds2bQU91bM36garf1k4sBTcRBFgxarfwE91keB2dtUxcn8sME9nbs36gm9rduC5R78,0mAUyf0d14BME91kbB36QLc12AB2dgyjqkHEUeMNT9157eQU9RMFT8s78C8neuixLc1zk4AtUxc1z8Mmt8re0fn8sWhLyc1bH36pl8neu,Kxycsw,iAxc1420l,K8ren8NS9n81bs36hc0vz8WmYzqkmhyv2WBHhyVOHXkJoSjIwSjIuSvz4WASVZIAXZ6skmSj6oFXzOmplvcsW46D61csk46plv8WBFDqoF,tarvk8WBH,tyjkqoHhGqkN,tmvZ8sWmhVZqskmpc0vZ8WAXZqkAplbnImASbn6skwSbn6skuSVOwSVOupGONSbn6soFpyVkJk5aSj6sk78YJkuDkIP5aYOuhvzk4WBAhVzk416oA,tyjkJ265a,,0mxyjk41q53sYzIHmPXkqowXkqouhyVqoHFYz6omFhb0e1zqkmNSyVIP78YJ20klpyVOHwYk620olpc0vz8WBmFXzqomFpG61ckH38PhyjIP78Yz620kmlDkImLDzINUhGIuNDzIA78hb0e1ZIANYkqk366chG6oFNXkJkP5ahVZ6somFSb0e1620kNlhVk41qomADzIFLXkqso78pGqoFNXzkImP5a,tyjk620oHlhG620kNlXzqskm78,tjZqskHmPYqouFD6sku78YzqkNU,tjZqsomF')[v[0]]]
 for o in range(len(v)-1):c=g(c,v)
 return c[-1]

You can test that f(2**32 - 1) runs almost instantly

I used this paper on a method for computing it. With this method there is a massive chunk of data for the pre-determined values for n from 78 to 334 without the even numbers after 168. I wanted to turn this data into something small and I didn't know any good compression algorithms so I made my own.

The way I compressed it was by having a list of string replace rules. I created a method which found the string replace rule which would cut down the most content over all taking into account defining the rule. I then recursively applied this until I could create no more rules (I used characters g-z and A-Z). The string I made to replace with was a comma separated list of the hex values for each of the numbers. In retrospect, converting them to their hex values may not have been the wisest choice, it would probably be shorter to leave them in decimal, since having hex would only save for the 3 digit numbers but would add a 0 for single digit numbers.

The line where I set c you can see the list of replace rules and the text which it runs it on. The rules need to be applied in reverse as well because some rules include characters created from other rules.

There are also numerous places in this code where I could probably cut down on syntax, such as turning the list of lists into a single list and then using a different method to access the rules to replace the text with

Answer 3

Haskell, 93 bytes

import Data.List
import Data.Ratio
p n=[x|x<-subsequences[2..n],sum x==n,1==sum(map(1%)x)]!!0

Horribly slow¹ but it runs in constant memory. Trivial solution: check all subsequences of [2..n] for sum and sum of reciprocals.

Returning all solutions instead of one is 3 bytes shorter: just remove the !!0 (beware: the running time will always be off the charts).

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¹ The running time depends on how early the result appears in the list of subsequences. Haskell's laziness stops the search if the first match is found. When compiled, p 89 (result: [3,4,6,9,18,21,28]) runs on my (4 year old) laptop in 35s. Other values, even smaller ones, can take hours.

Answer 4

Julia, 77 bytes

n->collect(filter(i->i==∪(i)&&sum(j->Rational(1,j),i)==1,partitions(n)))[1]

This is an inefficient lambda function that accepts an integer and returns an integer array. To call it, assign it to a variable.

We get the partitions of the integer using partitions. We then filter the set of partitions to only those with unique elements whose reciprocals sum to 1. To ensure no roundoff error occurs, we use Julia's Rational type to construct the reciprocals. filter returns an iterator, so we have to collect it into an array. This gives us an array of arrays (with only a single element), so we can get the first using [1].

Now, when I say inefficient, I mean it. Running this for n = 80 takes 39.113 seconds on my computer and allocates 13.759 GB of memory.