decompwlj 3D

Number of elements in the set {(x,y): 1 <= x,y <= n, gcd(x,y)=1}

A018805 on the OEIS · family summatory

Weight–level plate of Number of elements in the set {(x,y): 1 <= x,y <= n, gcd(x,y)=1}
Click the plate to explore it in 3-D. Weight k across, level L up, both on log scales: blue in the weight class (k ≤ L), orange in the level class (k > L). The dashed diagonal is k = L.
Open in the 3-D viewerA018805 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,995
Level class, k > L98,014 · 98.02 %
Weight class, k ≤ L1,981 · 1.98 %
Ties, k = L0
On the level line L = 110,774
Forced level, l ≤ d²79,671
Range of a(n)1 … 6,079,301,507
Range of the jump d2 … 199,980
Largest weight k, level L6,079,141,507, 126,463

18,412 different gaps occur, from 2 to 199,980; the level share is 98.02 %; 79.7 % of terms are forced level (l <= d^2); there are no ties.

The decomposition

Every term of a strictly increasing sequence is written a(n) = k(n)·L(n) + d(n): the jump d = a(n+1) − a(n), the weight k the least divisor of a − d greater than d, the level L = (a − d)/k. A term decomposes when a > 2d; it is in the level class when k > L and in the weight class when k ≤ L. See decompwlj.com and arXiv:0711.0865, or how it works, with worked examples and a live one.