Lets say you have numbers from 1 to 3, how many numbers can be made using powers of the numbers without the base repeating?
$1=1^1$
$2=2^1$
$3=3^1$
$4=3^1+1^1$
$5=3^1+2^1$
$6=3^1+2^1+1^1$
$7=3^1+2^2$
$8=2^3$
$9=3^2$
$10=3^2+1^1$
$11=3^2+2^1$
$12=3^2+2^1+1^1$
$13=3^2+2^2$
$14=3^2+2^2+1^1$
The answer is 14 because 15 cannot be made without 2 base numbers repeating
What I'm asking is. Using the sum of powers with the base of 1 to n, without a base repeat. How many numbers can you make?
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mathiscool
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Can the exponents be as large as $n$ or still just $3$? – Karl Aug 21 '23 at 05:14
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@Karl as large as n – mathiscool Aug 21 '23 at 05:14
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2So would it be accurate to rephrase your question as: What is the smallest positive integer that can not be written as a sum of positive integer powers of distinct elements of ${1,\dots,n}$? – Karl Aug 21 '23 at 05:17
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Your question is not totally clear. From your title, example and reply to Karl, you would appear to be asking for any given $\ n\ $ what is the largest number $\ x\ $ such that all natural numbers $\ k\ $ from $1$ to $x$ can be written in the form $$ k=\sum_{i=1}^ra_i^{b_i} $$ where $\ 1\le a_i\le n\ $, $\ 1\le b_i\le n\ $ and $\ a_i\ne a_j\ $ for $\ i\ne j\ $. Is that correct? – lonza leggiera Aug 21 '23 at 06:07
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Did you also try it for $n=4$ ? – Peter Aug 21 '23 at 13:43
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@peter Not yet. – mathiscool Aug 21 '23 at 20:50
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yes @lonzaleggiera – mathiscool Aug 21 '23 at 20:51
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yes, I will change the title to that. @Karl – mathiscool Aug 21 '23 at 20:53