Jelly, (削除) 12 (削除ここまで) 8 bytes
RÆṪSḤ’÷2
The following binary code works with this version of the Jelly interpreter.
0000000: 52 91 b0 53 aa b7 9a 8a R..S....
Idea
Clearly, the number of pairs (j, k) such that j ≤ k and j and k are co-prime equals the number of pairs (k, j) that satisfy the same conditions. Also, if j = k, j = 1 = k.
Thus, to count the number of co-prime pairs with coordinates between 1 and n, it suffices to calculate the amount m of pairs (j, k) withsuch that j ≤ k, then compute 2m - 1.
Finally, since Euler's totient function φ(k) yields the number integers between between 1 and k that are co-prime to k, we can calculate m as φ(1) + ... + φ(n).
Code
RÆṪSḤ’÷2 Input: n
R Yield [1, ..., n].
ÆṪ Apply Euler's totient function to each k in [1, ..., n].
S Compute the sum of all results.
Ḥ Multiply the result by 2.
’ Subtract 1.
÷2 Divide the result by n2.
Jelly, (削除) 12 (削除ここまで) 8 bytes
RÆṪSḤ’÷2
The following binary code works with this version of the Jelly interpreter.
0000000: 52 91 b0 53 aa b7 9a 8a R..S....
Idea
Clearly the number of pairs (j, k) such that j ≤ k and j and k are co-prime equals the number of pairs (k, j) that satisfy the same conditions. Also, if j = k, j = 1 = k.
Thus, to count the number of co-prime pairs with coordinates between 1 and n, it suffices to calculate the amount m of pairs (j, k) with j ≤ k, then compute 2m - 1.
Finally, since Euler's totient function φ(k) yields the number integers between between 1 and k that are co-prime to k, we can calculate m as φ(1) + ... φ(n).
Code
RÆṪSḤ’÷2 Input: n
R Yield [1, ..., n].
ÆṪ Apply Euler's totient function to each k in [1, ..., n].
S Compute the sum of all results.
Ḥ Multiply the result by 2.
’ Subtract 1.
÷2 Divide the result by n2.
Jelly, (削除) 12 (削除ここまで) 8 bytes
RÆṪSḤ’÷2
The following binary code works with this version of the Jelly interpreter.
0000000: 52 91 b0 53 aa b7 9a 8a R..S....
Idea
Clearly, the number of pairs (j, k) such that j ≤ k and j and k are co-prime equals the number of pairs (k, j) that satisfy the same conditions. Also, if j = k, j = 1 = k.
Thus, to count the number of co-prime pairs with coordinates between 1 and n, it suffices to calculate the amount m of pairs (j, k) such that j ≤ k, then compute 2m - 1.
Finally, since Euler's totient function φ(k) yields the number integers between between 1 and k that are co-prime to k, we can calculate m as φ(1) + ... + φ(n).
Code
RÆṪSḤ’÷2 Input: n
R Yield [1, ..., n].
ÆṪ Apply Euler's totient function to each k in [1, ..., n].
S Compute the sum of all results.
Ḥ Multiply the result by 2.
’ Subtract 1.
÷2 Divide the result by n2.
Jelly, (削除) 12 (削除ここまで) 8 bytes
RÆṪSḤ’÷2
The following binary code works with this version of the Jelly interpreter.
0000000: 52 91 b0 53 aa b7 9a 8a R..S....
How it worksIdea
Clearly the number of pairs (j, k) such that j ≤ k and j and k are co-prime equals the number of pairs (k, j) that satisfy the same conditions. Also, if j = k, j = 1 = k.
Thus, to count the number of co-prime pairs with coordinates between 1 and n, it suffices to calculate the amount m of pairs (j, k) with j ≤ k, then compute 2m - 1.
Finally, since Euler's totient function φ(k) yields the number integers between between 1 and k that are co-prime to k, we can calculate m as φ(1) + ... φ(n).
Code
RÆṪSḤ’÷2 Input: n
R Yield [1, ..., n].
ÆṪ Apply Euler's totient function to each k in [1, ..., n].
This yields the number of coprimes to k, less than or equal to k.
S Compute the sum of all results.
This yields the number of coprime pairs (j, k) such that j ≤ k.
Ḥ Multiply the result by 2.
This accounts for all (k, j) pairs, too.
’ Subtract 1 to account for counting (1, 1) twice.
÷2 Divide the result by n2.
Jelly, (削除) 12 (削除ここまで) 8 bytes
RÆṪSḤ’÷2
The following binary code works with this version of the Jelly interpreter.
0000000: 52 91 b0 53 aa b7 9a 8a R..S....
How it works
RÆṪSḤ’÷2 Input: n
R Yield [1, ..., n].
ÆṪ Apply Euler's totient function to each k in [1, ..., n].
This yields the number of coprimes to k, less than or equal to k.
S Compute the sum of all results.
This yields the number of coprime pairs (j, k) such that j ≤ k.
Ḥ Multiply the result by 2.
This accounts for all (k, j) pairs, too.
’ Subtract 1 to account for counting (1, 1) twice.
÷2 Divide the result by n2.
Jelly, (削除) 12 (削除ここまで) 8 bytes
RÆṪSḤ’÷2
The following binary code works with this version of the Jelly interpreter.
0000000: 52 91 b0 53 aa b7 9a 8a R..S....
Idea
Clearly the number of pairs (j, k) such that j ≤ k and j and k are co-prime equals the number of pairs (k, j) that satisfy the same conditions. Also, if j = k, j = 1 = k.
Thus, to count the number of co-prime pairs with coordinates between 1 and n, it suffices to calculate the amount m of pairs (j, k) with j ≤ k, then compute 2m - 1.
Finally, since Euler's totient function φ(k) yields the number integers between between 1 and k that are co-prime to k, we can calculate m as φ(1) + ... φ(n).
Code
RÆṪSḤ’÷2 Input: n
R Yield [1, ..., n].
ÆṪ Apply Euler's totient function to each k in [1, ..., n].
S Compute the sum of all results.
Ḥ Multiply the result by 2.
’ Subtract 1.
÷2 Divide the result by n2.
JellyJelly,(削除) 12 (削除ここまで) 8 bytes
RÆṪSḤ’÷2
Rest to be updated. Try it online!
Jelly , 12 bytes
Rb‘xgR=1SS÷2
The following binary code works with this version of the Jelly interpreter.
0000000: 52 62 b6 78 67 52 3d91 31b0 53 53aa b7 9a 8a RbR.xgR=1SS.S....
How it works
Rb‘xgR=1SS÷2RÆṪSḤ’÷2 Input: n
R Yield [1, ..., n].
b‘ÆṪ Apply Euler's Converttotient eachfunction to baseeach n+1.k Yieldsin [[1][1, ..., [n]]n].
x This Repeatyields eachthe nnumber times.of Yieldscoprimes [[1,to ...k, 1],less ...,than [n,or ...,equal n]]to k.
gRS TakeCompute GCDthe withsum [1,of ...,all n]results.
This yields allthe n2number GCDsof ascoprime apairs 10x10(j, arrayk) such that j ≤ k.
Ḥ =1 CheckMultiply forthe equalityresult withby 12.
S This Computeaccounts thefor sumall of(k, allj) columnspairs, too.
’ S Subtract 1 Computeto theaccount sumfor ofcounting all(1, sums1) twice.
÷2 ÷2 Divide the result by n2.
Jelly, 8 bytes
RÆṪSḤ’÷2
Rest to be updated. Try it online!
Jelly , 12 bytes
Rb‘xgR=1SS÷2
The following binary code works with this version of the Jelly interpreter.
0000000: 52 62 b6 78 67 52 3d 31 53 53 9a 8a Rb.xgR=1SS..
How it works
Rb‘xgR=1SS÷2 Input: n
R Yield [1, ..., n].
b‘ Convert each to base n+1. Yields [[1], ..., [n]].
x Repeat each n times. Yields [[1, ..., 1], ..., [n, ..., n]].
gR Take GCD with [1, ..., n].
This yields all n2 GCDs as a 10x10 array.
=1 Check for equality with 1.
S Compute the sum of all columns.
S Compute the sum of all sums.
÷2 Divide the result by n2.
Jelly,(削除) 12 (削除ここまで) 8 bytes
RÆṪSḤ’÷2
The following binary code works with this version of the Jelly interpreter.
0000000: 52 91 b0 53 aa b7 9a 8a R..S....
How it works
RÆṪSḤ’÷2 Input: n
R Yield [1, ..., n].
ÆṪ Apply Euler's totient function to each k in [1, ..., n].
This yields the number of coprimes to k, less than or equal to k.
S Compute the sum of all results.
This yields the number of coprime pairs (j, k) such that j ≤ k.
Ḥ Multiply the result by 2.
This accounts for all (k, j) pairs, too.
’ Subtract 1 to account for counting (1, 1) twice.
÷2 Divide the result by n2.