Carmichael lambda function: Difference between revisions
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→{{header|Julia}}: typo
(Created page with "{{draft task}} {{task|Prime Numbers}} ;Background The '''Carmichael function''', or '''Carmichael lambda function''', is a function in numeric theory. The Carmichael lambda {{math | ''λ''(''n'')}} of a positive integer '''n''' is the smallest positive integer '''n''' such that :<math>a^m \equiv 1 \pmod{n}</math> holds for every integer coprime to '''n'''. The Carmichael lambda function can be iterated, that is, called repeatedly on its result. If this iteration is p...") |
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{{draft task|Prime Numbers}}
;Background
The '''Carmichael function''', or '''Carmichael lambda function''', is a function in
:<math>a^m \equiv 1 \pmod{n}</math>
holds for every integer coprime to '''n'''.
Line 28 ⟶ 27:
:* [[wp:Carmichael_function|Wikipedia entry for Carmichael function]]
:* [[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}}==
{{trans|Python}}
<syntaxhighlight lang="Mathematica">
IteratedToOne[i_] := Module[{k = 0, iter = i},
While[iter > 1, iter = CarmichaelLambda[iter];
k++;];
k]
Print["Listing of (n, lambda(n), k for iteration to 1) for integers from 1 to 25:"];
Do[lam = CarmichaelLambda[i];
k = IteratedToOne[i];
If[Mod[i, 5] == 0, Print[{i, lam, k}], Print[{i, lam, k}, " "]], {i, 1, 25}]
upTo = 20;
maxToTest = 100000000000;
firsts = Table[0, {upTo + 1}];
firsts[[1]] = 1;
Print["\nIterations to 1 n lambda(n)\n=================================="];
Print[StringForm["`` `` ``", 0, 1, 1]];
Do[n = IteratedToOne[i];
If[0 < n <= upTo && firsts[[n + 1]] == 0, firsts[[n + 1]] = i;
j = If[n < 1, 0, CarmichaelLambda[i]];
Print[StringForm["`` `` ``", n, i, j]];
If[n >= upTo, Break[]];], {i, 2, maxToTest}]
</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
</pre>
=={{header|Phix}}==
<syntaxhighlight lang="Phix">
with javascript_semantics
function lambda(sequence pk)
integer {p,k} = pk, res = phi(power(p,k))
return iff(p=2 and k>=3 ? res/2 : res)
end function
function reduced_totient(integer n)
sequence p = prime_powers(n)
for i,pk in p do
p[i] = lambda(pk)
end for
return lcm(p)
-- alt: neater but ~2* slower
-- return lcm(apply(prime_powers(n),lambda))
end function
function k_iter(integer i)
return iff(i>1?1+k_iter(reduced_totient(i)):0)
end function
function ident(integer i) return i end function
sequence n = tagset(25)
for i,d in {"n","eulers_totient","reduced_totient","k"} do
sequence r = apply(n,{ident,phi,reduced_totient,k_iter}[i])
printf(1,"%17s: %s\n",{d,join(r," ",fmt:="%2d")})
end for
atom t0 = time()
printf(1,"\n k i lambda(i)\n")
printf(1, "==========================\n")
integer i = 1
for k=0 to iff(platform()=JS?12:15) do -- (keep JS < 3s)
while k_iter(i)!=k do i += 1 end while
printf(1,"%2d %,9d %,9d\n",{k,i,reduced_totient(i)})
end for
?elapsed(time()-t0)
</syntaxhighlight>
{{out}}
<pre>
n: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
eulers_totient: 1 1 2 2 4 2 6 4 6 4 10 4 12 6 8 8 16 6 18 8 12 10 22 8 20
reduced_totient: 1 1 2 2 4 2 6 2 6 4 10 2 12 6 4 4 16 6 18 4 6 10 22 2 20
k: 0 1 2 2 3 2 3 2 3 3 4 2 3 3 3 3 4 3 4 3 3 4 5 2 4
k i lambda(i)
==========================
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 1,439 1,438
10 2,879 2,878
11 34,549 34,548
12 138,197 138,196
13 531,441 354,294
14 1,594,323 1,062,882
15 4,782,969 3,188,646
"2 minutes and 27s"
</pre>
=== stretch ===
<pre>
16 14,348,907 9,565,938
17 43,046,721 28,697,814
</pre>
I estimate that took ~47mins, at which point I killed it.<br>
The Python code took 13mins 58 secs to get to 15, on the same box.
=={{header|Python}}==
Python has the Carmichael function in the SymPy library, where it is called reduced_totient.
<syntaxhighlight lang="Python">from sympy import reduced_totient
def iterated_to_one(i):
""" return k for the k-fold iterated lambda function where k is the first time iteration reaches 1 """
k = 0
while i > 1:
i = reduced_totient(i)
k += 1
return k
if __name__ == "__main__":
print("Listing of (n, lambda(n), k for iteration to 1) for integers from 1 to 25:")
for i in range(1, 26):
lam = reduced_totient(i)
k = iterated_to_one(i)
print(f'({i}, {lam}, {k})', end = '\n' if i % 5 == 0 else ' ')
UP_TO = 20
MAX_TO_TEST = 100_000_000_000
FIRSTS = [0] * (UP_TO + 1)
FIRSTS[0] = 1
print('\nIterations to 1 n lambda(n)\n==================================')
print(' 0 1 1')
for i in range(2, MAX_TO_TEST):
n = iterated_to_one(i)
if 0 < n <= UP_TO and FIRSTS[n] == 0:
FIRSTS[n] = i
j = 0 if n < 1 else int(reduced_totient(i))
print(f"{n:>4}{i:>17}{j:>11}")
if n >= UP_TO:
break
</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
</pre>
=={{header|Raku}}==
{{trans|Wren}}
<syntaxhighlight lang="raku" line># 20240228 Raku programming solution
use Prime::Factor;
sub phi(Int $p, Int $r) { return $p**($r - 1) * ($p - 1) }
sub CarmichaelLambda(Int $n) {
state %cache = 1 => 1, 2 => 1, 4 => 2;
sub CarmichaelHelper(Int $p, Int $r) {
my Int $n = $p ** $r;
return %cache{$n} if %cache{$n}:exists;
return %cache{$n} = $p > 2 ?? phi($p, $r) !! phi($p, $r - 1)
}
if $n < 1 { die "'n' must be a positive integer." }
return %cache{$n} if %cache{$n}:exists;
if ( my %pps = prime-factors($n).Bag ).elems == 1 {
my ($p, $r) = %pps.kv>>.Int;
return %cache{$n} = $p > 2 ?? phi($p, $r) !! phi($p, $r - 1)
}
return [lcm] %pps.kv.map: -> $k, $v { CarmichaelHelper($k.Int, $v) }
}
sub iteratedToOne($i is copy) {
my $k = 0;
while $i > 1 { $i = CarmichaelLambda($i) andthen $k++ }
return $k
}
say " n λ k";
say "----------";
for 1..25 -> $n {
printf "%2d %2d %2d\n", $n, CarmichaelLambda($n), iteratedToOne($n)
}
say "\nIterations to 1 i lambda(i)";
say "=====================================";
say " 0 1 1";
my ($maxI, $maxN) = 5e6, 10; # for N=15, takes around 47 minutes with an i5-10500T
my @found = True, |( False xx $maxN );
for 1 .. $maxI -> $i {
unless @found[ my $n = iteratedToOne($i) ] {
printf "%4d %18d %12d\n", $n, $i, CarmichaelLambda($i);
@found[$n] = True andthen ( last if $n == $maxN )
}
}</syntaxhighlight>
{{out}} Same as Wren example .. and it is probably 30X times slower 😅
=={{header|Wren}}==
{{libheader|Wren-math}}
{{libheader|Wren-fmt}}
Takes about 88 seconds on my machine (Core i7) to get up to n = 15 for the third part of the task, so haven't attempted the stretch goal.
<syntaxhighlight lang="wren">import "./math" for Int
import "./fmt" for Fmt
var primePows = Fn.new { |n|
var factPows = []
var pf = Int.primeFactors(n)
var currFact = pf[0]
var count = 1
for (fact in pf.skip(1)) {
if (fact != currFact) {
factPows.add([currFact, count])
currFact = fact
count = 1
} else {
count = count + 1
}
}
factPows.add([currFact, count])
return factPows
}
var phi = Fn.new { |p, r| p.pow(r-1) * (p - 1) }
var cache = { 1: 1, 2: phi.call(2, 1), 4: phi.call(2, 2) }
var CarmichaelHelper = Fn.new { |p, r|
var n = p.pow(r)
if (cache.containsKey(n)) return cache[n]
if (p > 2) return cache[n] = phi.call(p, r)
return cache[n] = phi.call(p, r - 1)
}
var CarmichaelLambda = Fn.new { |n|
if (n < 1) Fiber.abort("'n' must be a positive integer.")
if (cache.containsKey(n)) return cache[n]
var pps = primePows.call(n)
if (pps.count == 1) {
var p = pps[0][0]
var r = pps[0][1]
if (p > 2) return cache[n] = phi.call(p, r)
return cache[n] = phi.call(p, r - 1)
}
var a = []
for (pp in pps) a.add(CarmichaelHelper.call(pp[0], pp[1]))
return cache[n] = Int.lcm(a)
}
var iteratedToOne = Fn.new { |i|
var k = 0
while (i > 1) {
i = CarmichaelLambda.call(i)
k = k + 1
}
return k
}
System.print(" n λ k")
System.print("----------")
for (n in 1..25) {
var lambda = CarmichaelLambda.call(n)
var k = iteratedToOne.call(n)
Fmt.print("$2d $2d $2d", n, lambda, k)
}
System.print("\nIterations to 1 i lambda(i)")
System.print("=====================================")
System.print(" 0 1 1")
var maxI = 5 * 1e6
var maxN = 15
var found = List.filled(maxN + 1, false)
found[0] = true
var i = 1
while (i <= maxI) {
var n = iteratedToOne.call(i)
if (!found[n]) {
found[n] = true
var lambda = CarmichaelLambda.call(i)
Fmt.print("$4d $,18d $,12d", n, i, lambda)
if (n == maxN) break
}
i = i + 1
}</syntaxhighlight>
{{out}}
<pre>
n λ k
----------
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 i lambda(i)
=====================================
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 1,439 1,438
10 2,879 2,878
11 34,549 34,548
12 138,197 138,196
13 531,441 354,294
14 1,594,323 1,062,882
15 4,782,969 3,188,646
</pre>
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