decompwlj 3D

Semiprimes of the form k^2+1

A144255 on the OEIS · family multiplicative

Weight–level plate of Semiprimes of the form k^2+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 viewerA144255 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,996
Level class, k > L99,996 · 100.00 %
Weight class, k ≤ L0 · 0.00 %
Ties, k = L0
On the level line L = 16,160
Forced level, l ≤ d²99,996
Range of a(n)10 … 222,238,702,085
Range of the jump d16 … 31,168,059
Largest weight k, level L222,065,252,573, 234,057

99,158 different gaps occur, from 16 to 31,168,059; every decomposable term is forced level (l <= d^2).

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.