perplexus.info :: Just Math : Dudeney minus one: Dudezero

Dudeney minus one: Dudezero (Posted on 2022-02-01)

Sloane's A046459 has an article on Dudeney numbers, which are integers equal to sum of the digits of their cubes.
For example: 18³=5832, and 5+8+3+2=18. Accordingly, 18 is a Dudeney number.
The said article lists 0,1,8,18,26, and 27 as all possible Dudeney numbers.

A Dudezero number is a base 10 nonnegative integer the digits of whose cubes are each reduced by 1 and the said integer is equal to the sum of transformed digits.
For example, 5 is a Dudezero number, since: 5³=125, and:
(1-1)+(2-1)+(5-1)=5

Determine all possible Dudezero numbers.

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)

computer solution (spoiler)

| Comment 1 of 6

for n=1:10000

cube=n^3;

cubestr=char(string(cube));

tot=sod(cube)-length(cubestr);

if tot==n

disp([n cube])

end

finds the given 125 and only two others:

Testing to 10000 was overkill, but clearly any larger number would result in 12-digit or higher numbers, which the sum of 12 single-digit numbers couldn't reach.

The function sod() is in the path and defined as:

function sd = sod(n)

s=0;

dgts=num2str(n) ;

l=length(dgts) ;

for i=1:l

s=s+str2double(dgts(i));

end

sd=s;

end

Posted by Charlie on 2022-02-01 07:56:48

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