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.
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 Brauer 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, 191, 382 , 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)
EchoLisp
<lang scheme>
- 2^n
(define exp2 (build-vector 32 (lambda(i)(expt 2 i))))
- counters and results
(define-values (*minlg* *counts* *chains* *calls*) '(0 null null 0))
(define (register-hit chain lg ) (define idx (if (brauer? chain lg) 0 1))
(when (< lg *minlg*)
(set! *counts* (make-vector 2 0))
(set! *chains* (make-vector 2 ""))
(set! *minlg* lg))
(vector+= *counts* idx 1)
(vector-set! *chains* idx (vector->list chain)))
- is chain a brauer chain ?
(define (brauer? chain lg)
(for [(i (in-range 1 lg))]
#:break (not (vector-search* (- [chain i] [chain (1- i)]) chain)) => #f
#t))
- all min chains to target n (brute force)
(define (chains n chain lg (a) (top) (tops null)) (++ *calls*) (set! top [chain lg])
(cond
[(> lg *minlg*) #f] ;; too long
[(= n top) (register-hit chain lg)] ;; hit
[(< n top) #f] ;; too big
[(and (< *minlg* 32) (< (* top [exp2 (- *minlg* lg)]) n)) #f] ;; too small
[else
(for* ([i (in-range lg -1 -1)] [j (in-range lg (1- i) -1)])
(set! a (+ [chain i] [chain j]))
#:continue (<= a top) ;; increasing sequence
#:continue (memq a tops) ;; prevent duplicates
(set! tops (cons a tops))
(vector-push chain a)
(chains n chain (1+ lg))
(vector-pop chain))]))
(define (task n)
(set!-values (*minlg* *calls*) '(Infinity 0 ))
(chains n (make-vector 1 1) 0)
(printf "L(%d) = %d - brauer-chains: %d non-brauer: %d chains: %a %a "
n *minlg* [*counts* 0] [*counts* 1] [*chains* 0] [*chains* 1]))
</lang>
- Output:
(for-each task {7 14 21 29 32 42 64})
L(7) = 4 - brauer-chains: 5 non-brauer: 0 chains: (1 2 3 4 7)
L(14) = 5 - brauer-chains: 14 non-brauer: 0 chains: (1 2 3 4 7 14)
L(21) = 6 - brauer-chains: 26 non-brauer: 3 chains: (1 2 3 4 7 14 21) (1 2 4 5 8 13 21)
L(29) = 7 - brauer-chains: 114 non-brauer: 18 chains: (1 2 3 4 7 11 18 29) (1 2 3 6 9 11 18 29)
L(32) = 5 - brauer-chains: 1 non-brauer: 0 chains: (1 2 4 8 16 32)
L(42) = 7 - brauer-chains: 78 non-brauer: 6 chains: (1 2 3 4 7 14 21 42) (1 2 4 5 8 13 21 42)
L(64) = 6 - brauer-chains: 1 non-brauer: 0 chains: (1 2 4 8 16 32 64)
;; a few extras
(task 47)
L(47) = 8 - brauer-chains: 183 non-brauer: 37 chains: (1 2 3 4 7 10 20 27 47) (1 2 3 5 7 14 19 28 47)
(task 79)
L(79) = 9 - brauer-chains: 492 non-brauer: 129 chains: (1 2 3 4 7 9 18 36 43 79) (1 2 3 5 7 12 24 31 48 79)
Scala
Following Scala implementation finds number of minimum length Brauer chains and corresponding length. <lang Scala> object chains{
def check_seq(pos:Int,seq:List[Int],n:Int,min_len:Int):(Int,Int) = {
if(pos>min_len || seq(0)>n) (min_len,0)
else if(seq(0) == n) (pos,1)
else if(pos<min_len) try_perm(0,pos,seq,n,min_len)
else (min_len,0)
}
def try_perm(i:Int,pos:Int,seq:List[Int],n:Int,min_len:Int):(Int,Int) = {
if(i>pos) return (min_len,0)
val res1 = check_seq(pos+1,seq(0)+seq(i) :: seq,n,min_len)
val res2 = try_perm(i+1,pos,seq,n,res1._1)
if(res2._1 < res1._1) res2
else if(res2._1 == res1._1) (res2._1,res1._2 + res2._2)
else {
println("Try_perm exception")
(0,0)
}
}
val init_try_perm = (x:Int) => try_perm(0,0,List[Int](1),x,10)
def find_brauer(num:Int): Unit = {
val res = init_try_perm(num)
println()
println("N = %d".format(num))
println("Minimum length of chains: L(n)= " + res._1 + f"\nNumber of minimum length Brauer chains: " + res._2)
}
def main(args:Array[String]) :Unit = {
val nums = List(7,14,21,29,32,42,64)
for (i <- nums) find_brauer(i)
}
} </lang>
N = 7 Minimum length of chains: L(n)= 4 Number of minimum length Brauer chains: 5 N = 14 Minimum length of chains: L(n)= 5 Number of minimum length Brauer chains: 14 N = 21 Minimum length of chains: L(n)= 6 Number of minimum length Brauer chains: 26 N = 29 Minimum length of chains: L(n)= 7 Number of minimum length Brauer chains: 114 N = 32 Minimum length of chains: L(n)= 5 Number of minimum length Brauer chains: 1 N = 42 Minimum length of chains: L(n)= 7 Number of minimum length Brauer chains: 78 N = 64 Minimum length of chains: L(n)= 6 Number of minimum length Brauer chains: 1 N = 47 Minimum length of chains: L(n)= 8 Number of minimum length Brauer chains: 183 N = 79 Minimum length of chains: L(n)= 9 Number of minimum length Brauer chains: 492 N = 191 Minimum length of chains: L(n)= 11 Number of minimum length Brauer chains: 7172 N = 191 Minimum length of chains: L(n)= 11 Number of minimum length Brauer chains: 7172 N = 382 Minimum length of chains: L(n)= 11 Number of minimum length Brauer chains: 4 N = 379 Minimum length of chains: L(n)= 12 Number of minimum length Brauer chains: 6583
zkl
<lang zkl>var exp2=(32).pump(List,(2).pow), // 2^n, n=0..31
_minlg, _counts, _chains; // counters and results
fcn register_hit(chain,lg){ // save [upto 2] chains
idx:=(if(isBrauer(chain,lg)) 0 else 1);
if(lg<_minlg) _counts,_chains,_minlg=List(0,0), List("",""), lg;
_counts[idx]+=1;
_chains[idx]=chain.copy();
}
// is chain a brauer chain ?
fcn isBrauer(chain,lg){
foreach i in (lg){
if(not chain.holds(chain[i+1] - chain[i])) return(False);
}
True
}
// all min chains to target n (brute force)
fcn chains(n,chain,lg){
top,tops:=chain[lg], List();
if(lg>_minlg) {} // too long
else if(n==top) register_hit(chain,lg); // hit
else if(n<top) {} // too big
else if((_minlg<32) and (top*exp2[_minlg - lg]<n)){} // too small
else{
foreach i,j in ([lg..0,-1],[lg..i,-1]){
a:=chain[i] + chain[j];
if(a<=top) continue; // increasing sequence if(tops.holds(a)) continue; // prevent duplicates tops.append(a); chain.append(a); self.fcn(n,chain,lg+1); // recurse chain.pop();
} }
}</lang> <lang zkl>fcn task(n){
_minlg=(0).MAX;
chains(n,List(1),0);
println("L(%2d) = %d; Brauer-chains: %3d; non-brauer: %3d; chains: %s"
.fmt(n,_minlg,_counts.xplode(),_chains.filter()));
} T(7,14,21,29,32,42,64,47,79).apply2(task);</lang>
- Output:
L( 7) = 4; Brauer-chains: 5; non-brauer: 0; chains: L(L(1,2,3,4,7)) L(14) = 5; Brauer-chains: 14; non-brauer: 0; chains: L(L(1,2,3,4,7,14)) L(21) = 6; Brauer-chains: 26; non-brauer: 3; chains: L(L(1,2,3,4,7,14,21),L(1,2,4,5,8,13,21)) L(29) = 7; Brauer-chains: 114; non-brauer: 18; chains: L(L(1,2,3,4,7,11,18,29),L(1,2,3,6,9,11,18,29)) L(32) = 5; Brauer-chains: 1; non-brauer: 0; chains: L(L(1,2,4,8,16,32)) L(42) = 7; Brauer-chains: 78; non-brauer: 6; chains: L(L(1,2,3,4,7,14,21,42),L(1,2,4,5,8,13,21,42)) L(64) = 6; Brauer-chains: 1; non-brauer: 0; chains: L(L(1,2,4,8,16,32,64)) L(47) = 8; Brauer-chains: 183; non-brauer: 37; chains: L(L(1,2,3,4,7,10,20,27,47),L(1,2,3,5,7,14,19,28,47)) L(79) = 9; Brauer-chains: 492; non-brauer: 129; chains: L(L(1,2,3,4,7,9,18,36,43,79),L(1,2,3,5,7,12,24,31,48,79))