Numbers k such that the maximum exponent in its prime factorization equals the number of positive exponents (A051903(k) = A001221(k))

A212166 on the OEIS · family multiplicative

Weight–level plate of Numbers k such that the maximum exponent in its prime factorization equals the number of positive exponents (A051903(k) = A001221(k)) — 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 viewer A212166 on the OEIS

Terms	100,000 (n = 1 … 100,000)
Decomposable (a > 2d)	99,996
Level class, k > L	18,118 · 18.12 %
Weight class, k ≤ L	81,878 · 81.88 %
Ties, k = L	22
On the level line L = 1	8,554
Forced level, l ≤ d²	3
Range of a(n)	1 … 547,120
Range of the jump d	1 … 51
Largest weight k, level L	547,093, 273,505

47 different gaps occur, from 1 to 51; the level share is 18.12 %; L = 1 holds 47 % of the level class.

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.