Could it be that every positive integer is the sum of at most k such numbers, for some fixed k?
EDIT: No.
Proof. Fix a positive integer k. How many k-tuples of such numbers are there, whose sum is at most x? (For this to work, there would always need to be at least x of them.)
If (2^a_1 + 3^b_1 + 5^c_1 + 7^d_1) + ... + (2^a_k + 3^b_k + 5^c_k + 7^d_k) <= x, then all the exponents are at most log_2(x). So the number of all such sums is at most [log_2(x)]^(4k). And that is asymptotically less than x.
EDIT 2: But I'm sure the required k grows pretty slowly. This would be a workable technique for very large numbers indeed.
EDIT: No.
Proof. Fix a positive integer k. How many k-tuples of such numbers are there, whose sum is at most x? (For this to work, there would always need to be at least x of them.)
If (2^a_1 + 3^b_1 + 5^c_1 + 7^d_1) + ... + (2^a_k + 3^b_k + 5^c_k + 7^d_k) <= x, then all the exponents are at most log_2(x). So the number of all such sums is at most [log_2(x)]^(4k). And that is asymptotically less than x.
EDIT 2: But I'm sure the required k grows pretty slowly. This would be a workable technique for very large numbers indeed.