decompwlj 3D

Numbers whose sum of divisors is a triangular number

A045746 on the OEIS · family divisor functions

Weight–level plate of Numbers whose sum of divisors is a triangular number
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 viewerA045746 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,993
Level class, k > L62,998 · 63.00 %
Weight class, k ≤ L36,995 · 37.00 %
Ties, k = L4
On the level line L = 15,618
Forced level, l ≤ d²8,819
Range of a(n)1 … 420,028,981
Range of the jump d1 … 75,371
Largest weight k, level L420,023,561, 133,404,480

17,071 different gaps occur, from 1 to 75,371; the level share is 63.00 %; 8.8 % of terms are forced level (l <= d^2); 7 terms do not decompose.

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.