decompwlj 3D

Same numbers of prime factors of forms 4*k+1 and 4*k+3, counted with multiplicity

A072202 on the OEIS · family multiplicative

Weight–level plate of Same numbers of prime factors of forms 4*k+1 and 4*k+3, counted with multiplicity
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 viewerA072202 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,995
Level class, k > L15,988 · 15.99 %
Weight class, k ≤ L84,007 · 84.01 %
Ties, k = L24
On the level line L = 17,900
Forced level, l ≤ d²6
Range of a(n)1 … 461,422
Range of the jump d1 … 43
Largest weight k, level L461,381, 230,698

40 different gaps occur, from 1 to 43; the level share is 15.99 %; L = 1 holds 49 % 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.