decompwlj 3D

Primes having only {1, 3, 7, 9} as digits

A091633 on the OEIS · family primes

Weight–level plate of Primes having only {1, 3, 7, 9} as digits
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 viewerA091633 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,981
Level class, k > L29,229 · 29.23 %
Weight class, k ≤ L70,752 · 70.77 %
Ties, k = L2
On the level line L = 15,926
Forced level, l ≤ d²1,484
Range of a(n)3 … 3,911,131,199
Range of the jump d2 … 1,111,111,200
Largest weight k, level L3,911,131,067, 1,303,705,903

1,473 different gaps occur, from 2 to 1,111,111,200; the level share is 29.23 %; 1.5 % of terms are forced level (l <= d^2); 19 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.