three-base-thin

• Computes the three-base thin set E = {n : every base-3 digit <= 1, every base-5 digit <= 2, every base-7 digit <= 2} exactly, and the wider set with the base-7 bound at 3, which is {k : binomial(2k, k) is prime to 105}, A030979.
• Exact integer arithmetic throughout, num-bigint above 128 bits; no sampling, no fit.
• The walk enumerates the base-3 side, whose members are the subset sums of distinct powers of 3, from the top power down. Once the powers 3^k and above are chosen the remaining addition is at most (3^k - 1)/2, so with j least such that 5^j > (3^k - 1)/2 the high part floor(n / 5^j) of every n still reachable is one of two consecutive integers, and the branch dies when neither of them has all its base-5 digits inside the bound; base 7 cuts the same way. Membership in the base-3 set holds by construction and is never tested.
• Also prints the three dimensions of E, log_3 2, log_5 3 and log_7 3, the base-7 dimension log_7 4 of the wider set, the three pair budgets dim_i + dim_j - 1, the two triple budgets sum dim - 2, and the criterion A/(p-1) + B/(q-1) of Erdos, Graham, Ruzsa and Straus for each pair. A dimension prints to the nearest six places, a budget rounds up, an effective exponent truncates down.

VERBS

• budget prints the four dimensions, the three pair budgets, the triple budget for the thin set and for the wider one, and the five values of the infinitude criterion. Runtime under 0.01 s.
• control rebuilds the members below 10^8 twice, once by the pruned walk and once by scanning every integer against all three digit rules, at both base-7 bounds, and rebuilds the 23 terms A030979 publishes. Runtime 1.9 s.
• seven M prints the members of E below 7^M with the node count. Runtime under 0.01 s at M = 17.
• reach level prints the members of E below 3^level with the node count. Runtime 3.02 s at level 20000, 13.60 s at level 40000 and 61.48 s at level 80000, the last on about 3 GB, which is the wall: the stored powers cost Theta(level^2) bits and the node count grows near 21 level.
• wide level and ten X do the same at base-7 bound 3, below 3^level and below 10^X, printing the count, the largest member and the effective exponent log(count)/log(height). Runtime 0.09 s at X = 70 and 39.32 s at X = 140.

RUN

CARGO_BUILD_JOBS=4 cargo run --release -p three-base-thin -- budget
CARGO_BUILD_JOBS=4 cargo run --release -p three-base-thin -- control
CARGO_BUILD_JOBS=4 cargo run --release -p three-base-thin -- seven 17
CARGO_BUILD_JOBS=4 cargo run --release -p three-base-thin -- reach 80000
CARGO_BUILD_JOBS=4 cargo run --release -p three-base-thin -- ten 70
CARGO_BUILD_JOBS=4 cargo run --release -p three-base-thin -- ten 140

• CARGO_BUILD_JOBS=4 cargo test -p three-base-thin, 7 tests, 1.18 s after the build.

READS

• E below 7^17 = 232630513987207: 0, 1, 3186, 3187, 20007, 333 nodes.
• E below 3^80000, a height of 38170 decimal digits: the same five, 1710789 nodes.
• The wider set below 10^70: 1374 members, the length of the table A030979 calls complete to 10^70. Below 10^140: 216020 members.
• Dimensions 0.630930, 0.682606 and 0.564575 for E, and 0.712414 for the base-7 side of the wider set; pair budgets 0.313536, 0.195505, 0.247182; triple budgets -0.121889 thin and 0.025951 wide; criterion 1.000000 at (3, 5) and 0.833333 at (3, 7) and at (5, 7).
• Effective exponents of the wider set 0.044828 at 10^70 and 0.038103 at 10^140, both above 0.025951 and falling.

WITNESSES

• cobham.md:87 the three dimensions of E
• cobham.md:88 and cobham.md:89 the pair budgets and the triple budget
• cobham.md:91 and cobham.md:92 the infinitude criterion at each pair
• cobham.md:94 the members below 7^17
• cobham.md:95 the members below 3^80000
• cobham.md:96 the scan control below 10^8
• cobham.md:97 the fourth dimension log_7 4 = 0.712414 and the wider budget 0.025951
• cobham.md:99 the 23 published terms rebuilt, the 1374 below 10^70, the 216020 below 10^140 and the two effective exponents