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A373701
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 *.
0
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, 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, 9, 10, 12, 12, 8, 9, 8, 8, 9, 11, 13, 12, 9, 8, 8, 8, 9, 10
OFFSET
1,2
COMMENTS
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.
LINKS
EXAMPLE
| | rational | minimal expression | a(n) |
|---:|:-----------|:----------------------------|--------:|
| 1 | 1/1 | 1 | 1 |
| 2 | 1/2 | 1/(1+1) | 3 |
| 3 | 2/1 | 1+1 | 2 |
| 4 | 1/3 | 1/(1+1+1) | 4 |
| 5 | 3/1 | 1+1+1 | 3 |
| 6 | 1/4 | 1/(1+1+1+1) | 5 |
| 7 | 2/3 | (1+1)/(1+1+1) | 5 |
| 8 | 3/2 | 1+(1/(1+1)) | 4 |
| 9 | 4/1 | 1+1+1+1 | 4 |
| 10 | 1/5 | 1/(1+1+1+1+1) | 6 |
| 11 | 5/1 | 1+1+1+1+1 | 5 |
| 12 | 1/6 | 1/((1+1)*(1+1+1)) | 6 |
| 13 | 2/5 | (1+1)/(1+1+1+1+1) | 7 |
| 14 | 3/4 | (1+1+1)/(1+1+1+1) | 7 |
| 15 | 4/3 | 1+(1/(1+1+1)) | 5 |
| 16 | 5/2 | 1+1+(1/(1+1)) | 5 |
| 17 | 6/1 | (1+1)*(1+1+1) | 5 |
| 18 | 1/7 | 1/((1+1+1)*(1+1) +1) | 7 |
| 19 | 3/5 | (1+1+1)/(1+1+1+1+1) | 8 |
| 20 | 5/3 | 1+((1+1)/(1+1+1)) | 6 |
| 21 | 7/1 | (1+1)*(1+1+1)+1 | 6 |
| 22 | 1/8 | 1/((1+1)*(1+1)*(1+1)) | 7 |
| 23 | 2/7 | (1+1)/((1+1)*(1+1+1)+1) | 8 |
| 24 | 4/5 | (1+1+1+1)/(1+1+1+1+1) | 9 |
| 25 | 5/4 | 1+(1/(1+1+1+1)) | 6 |
| 26 | 7/2 | 1+1+1+(1/(1+1)) | 6 |
| 27 | 8/1 | (1+1)*(1+1)*(1+1) | 6 |
| 28 | 1/9 | 1/((1+1+1)*(1+1+1)) | 7 |
| 29 | 3/7 | (1+1+1)/((1+1+1)*(1+1) +1) | 9 |
| 30 | 7/3 | 1+1+(1/(1+1+1)) | 6 |
| 31 | 9/1 | (1+1+1)*(1+1+1) | 6 |
| 32 | 1/10 | 1/((1+1+1)*(1+1+1)+1) | 8 |
| 33 | 2/9 | (1+1)/((1+1+1)*(1+1+1)) | 8 |
| 34 | 3/8 | (1+1+1)/((1+1)*(1+1)*(1+1)) | 9 |
| 35 | 4/7 | (1+1+1+1)/((1+1)*(1+1+1)+1) | 10 |
| 36 | 5/6 | (1/(1+1))+(1/(1+1+1)) | 7 |
| 37 | 6/5 | 1+(1/(1+1+1+1+1)) | 7 |
CROSSREFS
Cf. A005245 (Mahler-Popken complexity).
Ordering used: A020652 (Cantor numerators), A020653 (Cantor denominators).
Sequence in context: A302391 A140114 A243852 * A025532 A329584 A335507
KEYWORD
nonn
AUTHOR
Adil Soubki, Jun 13 2024
STATUS
approved