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%I #13 Jul 06 2024 13:45:51
%S 1,3,2,4,3,5,5,4,4,6,5,6,7,7,5,5,5,7,8,6,6,7,8,9,6,6,6,7,9,6,6,8,8,9,
%T 10,7,7,8,7,7,7,9,11,8,8,8,10,10,10,10,11,7,9,7,7,8,7,9,11,11,10,8,8,
%U 9,10,12,12,8,9,8,8,9,11,13,12,9,8,8,8,9,10
%N Extension of Mahler-Popken complexity to the rationals. The minimal number of 1's required to build the n-th positive rational in the Cantor ordering using only +, /, and *.
%C Since we do not require that rationals with denominator 1 be written in the form p/q (i.e., we allow them to be written as p), this reduces to A005245 in the case where q = 1.
%H Dimitri Zucker, <a href="https://www.youtube.com/watch?v=RdnTi-2gahs">The Most Underrated Concept in Number Theory</a>, Combo Class Youtube video.
%e | | rational | minimal expression | a(n) |
%e |---:|:-----------|:----------------------------|--------:|
%e | 1 | 1/1 | 1 | 1 |
%e | 2 | 1/2 | 1/(1+1) | 3 |
%e | 3 | 2/1 | 1+1 | 2 |
%e | 4 | 1/3 | 1/(1+1+1) | 4 |
%e | 5 | 3/1 | 1+1+1 | 3 |
%e | 6 | 1/4 | 1/(1+1+1+1) | 5 |
%e | 7 | 2/3 | (1+1)/(1+1+1) | 5 |
%e | 8 | 3/2 | 1+(1/(1+1)) | 4 |
%e | 9 | 4/1 | 1+1+1+1 | 4 |
%e | 10 | 1/5 | 1/(1+1+1+1+1) | 6 |
%e | 11 | 5/1 | 1+1+1+1+1 | 5 |
%e | 12 | 1/6 | 1/((1+1)*(1+1+1)) | 6 |
%e | 13 | 2/5 | (1+1)/(1+1+1+1+1) | 7 |
%e | 14 | 3/4 | (1+1+1)/(1+1+1+1) | 7 |
%e | 15 | 4/3 | 1+(1/(1+1+1)) | 5 |
%e | 16 | 5/2 | 1+1+(1/(1+1)) | 5 |
%e | 17 | 6/1 | (1+1)*(1+1+1) | 5 |
%e | 18 | 1/7 | 1/((1+1+1)*(1+1) +1) | 7 |
%e | 19 | 3/5 | (1+1+1)/(1+1+1+1+1) | 8 |
%e | 20 | 5/3 | 1+((1+1)/(1+1+1)) | 6 |
%e | 21 | 7/1 | (1+1)*(1+1+1)+1 | 6 |
%e | 22 | 1/8 | 1/((1+1)*(1+1)*(1+1)) | 7 |
%e | 23 | 2/7 | (1+1)/((1+1)*(1+1+1)+1) | 8 |
%e | 24 | 4/5 | (1+1+1+1)/(1+1+1+1+1) | 9 |
%e | 25 | 5/4 | 1+(1/(1+1+1+1)) | 6 |
%e | 26 | 7/2 | 1+1+1+(1/(1+1)) | 6 |
%e | 27 | 8/1 | (1+1)*(1+1)*(1+1) | 6 |
%e | 28 | 1/9 | 1/((1+1+1)*(1+1+1)) | 7 |
%e | 29 | 3/7 | (1+1+1)/((1+1+1)*(1+1) +1) | 9 |
%e | 30 | 7/3 | 1+1+(1/(1+1+1)) | 6 |
%e | 31 | 9/1 | (1+1+1)*(1+1+1) | 6 |
%e | 32 | 1/10 | 1/((1+1+1)*(1+1+1)+1) | 8 |
%e | 33 | 2/9 | (1+1)/((1+1+1)*(1+1+1)) | 8 |
%e | 34 | 3/8 | (1+1+1)/((1+1)*(1+1)*(1+1)) | 9 |
%e | 35 | 4/7 | (1+1+1+1)/((1+1)*(1+1+1)+1) | 10 |
%e | 36 | 5/6 | (1/(1+1))+(1/(1+1+1)) | 7 |
%e | 37 | 6/5 | 1+(1/(1+1+1+1+1)) | 7 |
%Y Cf. A005245 (Mahler-Popken complexity).
%Y Ordering used: A020652 (Cantor numerators), A020653 (Cantor denominators).
%K nonn
%O 1,2
%A _Adil Soubki_, Jun 13 2024