Addition chains

An addition chain of length r for n is a sequence 1 = a(0) < a(1) < a(2) ... < a(r) = n , such as a(k) = a(i) + a(j) ( i < k and j < k , i may be = j) . Each member is the sum of two earlier members, not necessarily distincts.

Addition chains is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

A Brauer chain for n is an addition chain where a(k) = a(k-1) + a(j) with j < k. Each member uses the previous member as a summand.

We are interested in chains of minimal length l(n).

Task

For each n in {7,14,21,29,32,42,64} display the following : l(n), the count of Brauer chains of length l(n), an example of such a Brauser chain, the count of non-brauer chains of length l(n), an example of such a chain. (NB: counts may be 0 ).

Extra-credit: Same task for n in {47,79,379, 12509}

References

• OEIS sequences A079301, A079302. [1]
• Richard K. Guy - Unsolved problems in Number Theory - C6 - Addition chains.

Example

• minimal chain length l(19) = 6
• brauer-chains(19) : count = 31 Ex: #( 1 2 3 4 8 11 19)
• non-brauer-chains(19) : count = 2 Ex: #( 1 2 3 6 7 12 19)