Carmichael lambda function: Difference between revisions
m
→{{header|Julia}}: typo
(added Raku programming solution) |
m (→{{header|Julia}}: typo) |
||
| (One intermediate revision by the same user not shown) | |||
Line 28:
:* [[oeis:A181776|OEIS lambda(lamda(n)) function sequence]]
=={{header|Julia}}==
<syntaxhighlight lang="julia">using DelimitedFiles
using Primes
const maxcached = typemax(Int32)
const carmicache = zeros(Int32, maxcached) # NB: 8 gigabytes memory used for this size cache
const rlock = ReentrantLock()
""" Carmichael reduced totient function lambda(n) """
function lambda(n::Integer)
@assert n > 0
n <= maxcached && carmicache[n] > 0 && return Int(carmicache[n])
lam = 1
for (p, e) in factor(n)
if p == 2 && e > 2
lam = lcm(lam, 2^(e - 2))
else
lam = lcm(lam, (p - 1) * p^(e - 1))
end
end
if n <= maxcached
lock(rlock)
carmicache[n] = lam
unlock(rlock)
end
return lam
end
"""
Return k for the k-fold iterated lambda function, where k is count of iterations to 1
"""
function iterated_lambdas_to_one(i)
k = 0
while i > 1
i = lambda(i)
k += 1
end
return k
end
"""
Calculate values for the task. Due to a runtime of several days for a sequence length
of 25, the function switches to multithreading for values over about 2 billion. Note
that (empirically) adjacent pairs of values within the sequence above 100000 are less
than a factor of 6 apart, so each next large value is sought within that range.
"""
function print_iterated_lambdas(upto = 25)
println("Listing of (n, lambda(n), k for iteration to 1) for integers from 1 to 25:")
firsts = zeros(Int, upto)
if isfile("lambdas.txt")
precalc = readdlm("lambdas.txt", Int64)
if length(precalc) == upto
firsts .= precalc # restore work done if resterted (since it takes days)
end
end
for i = 1:25
lam = lambda(i)
k = iterated_lambdas_to_one(i)
print("($i, $lam, $k)", i % 5 == 0 ? "\n" : " ")
end
target = something(findlast(!iszero, firsts), 1)
if firsts[target] < maxcached ÷ 7 # get first ones with single thread
for i = 2:maxcached
n = iterated_lambdas_to_one(i)
if n <= upto && (firsts[n] == 0 || firsts[n] > i)
firsts[n] = i
end
end
end
target = something(findlast(!iszero, firsts), 1) + 1
while target <= upto # multithreaded for larger targets
startval = firsts[target - 1] + 1
for j in startval:startval:startval*6
j += iseven(j) # make odd
@Threads.threads for i in j:2:j+startval-1 # should not be even if i > 2
n = iterated_lambdas_to_one(i)
if n == target
if firsts[n] == 0 || firsts[n] > i
try
lock(rlock)
firsts[n] = i
writedlm("lambdas.txt", firsts) # save work done for restarts
unlock(rlock)
catch y
unlock(rlock)
@warn y
rethrow()
end
end
break
end
firsts[target] > 0 && firsts[target] < i && break
end
end
target += 1
end
println("\nIterations to 1 n lambda(n)\n=====================================")
println(" 0 1 1")
for n = 1:upto
println(lpad(n, 4) * lpad(firsts[n], 17) * lpad(lambda(firsts[n]), 14))
end
end
print_iterated_lambdas()
</syntaxhighlight>{{out}}
<pre>
Listing of (n, lambda(n), k for iteration to 1) for integers from 1 to 25:
(1, 1, 0) (2, 1, 1) (3, 2, 2) (4, 2, 2) (5, 4, 3)
(6, 2, 2) (7, 6, 3) (8, 2, 2) (9, 6, 3) (10, 4, 3)
(11, 10, 4) (12, 2, 2) (13, 12, 3) (14, 6, 3) (15, 4, 3)
(16, 4, 3) (17, 16, 4) (18, 6, 3) (19, 18, 4) (20, 4, 3)
(21, 6, 3) (22, 10, 4) (23, 22, 5) (24, 2, 2) (25, 20, 4)
Iterations to 1 n lambda(n)
=====================================
0 1 1
1 2 1
2 3 2
3 5 4
4 11 10
5 23 22
6 47 46
7 283 282
8 719 718
9 1439 1438
10 2879 2878
11 34549 34548
12 138197 138196
13 531441 354294
14 1594323 1062882
15 4782969 3188646
16 14348907 9565938
17 43046721 28697814
18 86093443 86093442
19 258280327 258280326
20 688747547 688747546
21 3486784401 2324522934
22 10460353203 6973568802
23 31381059609 20920706406
24 94143178827 62762119218
25 282429536481 188286357654
</pre>
=={{header|Mathematica}}|{{header|Wolfram Language}}==
| |||