decompwlj 3D

Primes which can be expressed as a sum of distinct powers of 3

A077717 on the OEIS · family primes

Weight–level plate of Primes which can be expressed as a sum of distinct powers of 3
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 viewerA077717 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,980
Level class, k > L38,135 · 38.14 %
Weight class, k ≤ L61,845 · 61.86 %
Ties, k = L0
On the level line L = 112,130
Forced level, l ≤ d²1,727
Range of a(n)3 … 11,811,362,257
Range of the jump d6 … 5,230,177,110
Largest weight k, level L11,811,362,023, 1,687,058,191

1,685 different gaps occur, from 6 to 5,230,177,110; the level share is 38.14 %; 1.7 % of terms are forced level (l <= d^2); L = 1 holds 32 % of the level class; there are no ties; 20 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.