Let k = p_1^e_1 p_2^e_2 p_3^e_3 ... be the prime factorization of n. Sequence gives k such that the sum of the numbers of 1's in the binary expansions of e_1, e_2, e_3, ... is odd

A000028 on the OEIS · family multiplicative

Weight–level plate of Let k = p_1^e_1 p_2^e_2 p_3^e_3 ... be the prime factorization of n. Sequence gives k such that the sum of the numbers of 1's in the binary expansions of e_1, e_2, e_3, ... is odd — 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 A000028 on the OEIS

Terms	100,000 (n = 1 … 100,000)
Decomposable (a > 2d)	99,999
Level class, k > L	12,744 · 12.74 %
Weight class, k ≤ L	87,255 · 87.26 %
Ties, k = L	40
On the level line L = 1	8,991
Forced level, l ≤ d²	3
Range of a(n)	2 … 200,014
Range of the jump d	1 … 17
Largest weight k, level L	199,999, 100,004

16 different gaps occur, from 1 to 17; the level share is 12.74 %; L = 1 holds 71 % 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.