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Greatest prime dividing the n-th cubefree number: Difference between revisions

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Greatest prime dividing the n-th cubefree number (view source)

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{{draft task}}
;Definitions
A cubefree number is a positive integer whose prime factorization does not contain any third (or higher) power factors. If follows that all primes are trivially cubefree and the first cubefree number is 1 because it has no prime factors.
Line 180:
The 1000000th term of a[n] is 1202057
The 10000000th term of a[n] is 1202057</pre>
 
=={{header|jq}}==
'''Works with jq, the C implementation of jq'''
 
'''Works with gojq, the Go implementation of jq'''
<syntaxhighlight lang="jq">
# The following may be omitted if using the C implementation of jq
def _nwise($n):
def n: if length <= $n then . else .[0:$n] , (.[$n:] | n) end;
n;
 
### Generic functions
def lpad($len): tostring | ($len - length) as $l | (" " * $l) + .;
 
# tabular print
def tprint($columns; $width):
reduce _nwise($columns) as $row ("";
. + ($row|map(lpad($width)) | join(" ")) + "\n" );
 
# like while/2 but emit the final term rather than the first one
def whilst(cond; update):
def _whilst:
if cond then update | (., _whilst) else empty end;
_whilst;
 
## Prime factors
 
# Emit an array of the prime factors of 'n' in order using a wheel with basis [2, 3, 5]
# e.g. 44 | primeFactors => [2,2,11]
def primeFactors:
def out($i): until (.n % $i != 0; .factors += [$i] | .n = ((.n/$i)|floor) );
if . < 2 then []
else [4, 2, 4, 2, 4, 6, 2, 6] as $inc
| { n: .,
factors: [] }
| out(2)
| out(3)
| out(5)
| .k = 7
| .i = 0
| until(.k * .k > .n;
if .n % .k == 0
then .factors += [.k]
| .n = ((.n/.k)|floor)
else .k += $inc[.i]
| .i = ((.i + 1) % 8)
end)
| if .n > 1 then .factors += [ .n ] else . end
| .factors
end;
 
### Cube-free numbers
# If cubefree then emit the largest prime factor, else emit null
def cubefree:
if . % 8 == 0 or . % 27 == 0 then false
else primeFactors as $factors
| ($factors|length) as $n
| {i: 2, cubeFree: true}
| until (.cubeFree == false or .i >= $n;
$factors[.i-2] as $f
| if $f == $factors[.i-1] and $f == $factors[.i]
then .cubeFree = false
else .i += 1
end)
| if .cubeFree then $factors[-1] else null end
end;
 
## The tasks
{ res: [1], # by convention
count: 1, # see the previous line
i: 2,
lim1: 100,
lim2: 1000,
max: 10000 }
| whilst (.count <= .max;
.emit = null
| (.i|cubefree) as $result
| if $result
then .count += 1
| if .count <= .lim1 then .res += [$result] end
| if .count == .lim1
then .emit = ["First \(.lim1) terms of a[n]:"]
| .emit += [.res | tprint(10; 3)]
elif .count == .lim2
then .lim2 *= 10
| .emit = ["The \(.count) term of a[n] is \($result)"]
end
end
| .i += 1
| if .i % 8 == 0 or .i % 27 == 0
then .i += 1
end
)
| select(.emit) | .emit[]
</syntaxhighlight>
{{output}}
<pre>
First 100 terms of a[n]:
1 2 3 2 5 3 7 3 5 11
3 13 7 5 17 3 19 5 7 11
23 5 13 7 29 5 31 11 17 7
3 37 19 13 41 7 43 11 5 23
47 7 5 17 13 53 11 19 29 59
5 61 31 7 13 11 67 17 23 7
71 73 37 5 19 11 13 79 41 83
7 17 43 29 89 5 13 23 31 47
19 97 7 11 5 101 17 103 7 53
107 109 11 37 113 19 23 29 13 59
 
The 1000 term of a[n] is 109
The 10000 term of a[n] is 101
The 100000 term of a[n] is 1693
The 1000000 term of a[n] is 1202057
</pre>
 
=={{header|Julia}}==
Line 553 ⟶ 667:
</pre>
===resursive alternative===
only counting til limit.Using Apéry's Constant, which is a quite good estimate.<br>Only checking powers of 10.Not willing to test prime factors > 2,642,246-> 0
<syntaxhighlight lang="pascal">
program CubeFree3;
Line 568 ⟶ 682:
const
//Apéry's Constant
Z3 : extended = 1.20205690315959428539973816151144999;
RezZ3 = 0.831907372580707468683126278821530734417;
{
limits :array[0..9] of UInt64 =
(1199,12019,120203,1202057,12020570,120205685,1202056919,
12020569022,120205690298,1202056903137);
}
 
type
Line 580 ⟶ 689:
tPrimes = array[tPrimeIdx] of Uint32;
tDl3 = UInt64;
tDelCubetPrmCubed = array[tPrimeIdx] of tDl3;
 
var
{$ALIGN 8}
SmallPrimes: tPrimes;
{$ALIGN 32}
DelCube : tDelCube;
PrmCubed : tPrmCubed;
 
procedure InitSmallPrimes;
Line 629 ⟶ 739:
end;
 
procedure InitDelCubeInitPrmCubed(var DC:tDelCubetPrmCubed);
var
i,q : Uint64;
Line 667 ⟶ 777:
end;
 
function highestDiv(n: uint64):Uint64;
procedure OutNum(lmt,n,CntDivs:Uint64);
//can test til 2642246^2 ~ 6,98E12
var
pr : Uint64;
i : integer;
begin
result := n;
writeln(Numb2Usa(lmt):26,'|',Numb2Usa(n):26,'|',Numb2Usa(CntDivs):10);
for i in tPrimeIdx do
begin
pr := Smallprimes[i];
if pr*pr>result then
BREAK;
while (result > pr) AND (result MOD pr = 0) do
result := result DIV pr;
end;
end;
 
procedure OutNum(lmt,n:Uint64);
var
MaxPrimeFac : Uint64;
begin
MaxPrimeFac := highestDiv(lmt);
if MaxPrimeFac > sqr(SmallPrimes[high(tPrimeIdx)]) then
MaxPrimeFac := 0;
writeln(Numb2Usa(lmt):26,'|',Numb2Usa(n):26,'|',Numb2Usa(MaxPrimeFac):15);
end;
//##########################################################################
var
cnt : Uint64Int64;
CntDivs : Uint32;
 
procedure check(lmt:Uint64;i:integer;flip :Boolean);
Line 682 ⟶ 813:
For i := i to high(tPrimeIdx) do
begin
p := DelCubePrmCubed[i];
if lmt < p then
BREAK;
inc(CntDivs);
p := lmt DIV p;
if flip then
Line 691 ⟶ 821:
else
cnt += p;
if p >= DelCubePrmCubed[i+1] then
check(p,i+1,not(flip));
end;
end;
 
function GetLmtfromCnt(inCnt:Uint64):Uint64;
procedure Checklmt(lmtIdx:Uint32);
var
limit : extended;
lmt: Uint64;
begin
limitresult := trunc(Z3*inCnt);
repeat
while LmtIdx>0 do
cnt := result;
Begin
check(result,0,true);
limit *= 10;
//new approximation
dec(LmtIdx);
inc(result,trunc(Z3*(inCnt-Cnt)));
end;
lmtuntil cnt := trunc(limit)inCnt;
//maybe lmt is not cubefree, like 1200 for cnt 1000
cnt := lmt;
//faster than checking for cubefree of lmt for big lmt
CntDivs := 0;
repeat
check(lmt,0,true);
dec(result);
OutNum(lmt,cnt,CntDivs);
cnt := result;
check(result,0,true);
until cnt < inCnt;
inc(result);
end;
//##########################################################################
 
var
T0,lmt:Int64;
i : integer;
Begin
InitSmallPrimes;
InitPrmCubed(PrmCubed);
InitDelCube(DelCube);
 
For i := 1 to 100 do
Begin
lmt := GetLmtfromCnt(i);
write(highestDiv(lmt):4);
if i mod 10 = 0 then
Writeln;
end;
Writeln;
 
Writeln('Tested with Apéry´s Constant approximation of ',Z3:17:15);
write(' ');
writeln('Limit | cube free numbers |countmax ofprim divisionsfactor');
T0 := GetTickCount64;
lmt := 1;
For i := 0 to 18 do
Begin
Checklmt(i);
OutNum(GetLmtfromCnt(lmt),lmt);
lmt *= 10;
end;
T0 := GetTickCount64-T0;
writeln(' runtime ',T0/1000:0:3,' s');
 
end.
end.</syntaxhighlight>
{{out|@home}}
<pre>
1 2 3 2 5 3 Limit |7 3 5 cube free numbers |count of divisions11
3 13 7 5 17 3 19 5 7 1| 1| 011
23 5 13 7 29 5 12| 31 11| 17 17
3 37 19 13 41 7 43 11 5 120| 101| 223
47 7 5 17 13 53 11 19 29 1,202| 1,002| 659
5 61 31 7 13 11 67 17 12,020|23 10,001| 147
71 73 37 5 19 11 13 79 120,205|41 100,001| 3083
7 17 43 29 89 5 13 1,202,056| 23 31 999,999| 6547
19 97 7 11 5 101 17 12,020,569|103 7 9,999,999| 14153
107 109 11 37 113 19 23 29 13 59
120,205,690| 100,000,004| 301
 
1,202,056,903| 999,999,986| 645
Tested with Apéry´s Constant approximation of 1.202056903159594
12,020,569,031| 10,000,000,008| 1,392
120,205,690,315 Limit | cube free numbers 100,000,000,014|max prim factor| 3,003divs
1,202,056,903,159| 1,000,000,000,020| 6,465 1| 0
12,020,569,031,595 11| 9,999,999,999,963 10| 13,924 11| 1
120,205,690,315,959 118| 100,000,000,000,027| 30,006 59| 2
1,202,056,903,159,594199| 1,000,000,000,000,087| 64,643 109| 6<
12,019| 10,000| 101| 14
12,020,569,031,595,942| 9,999,999,999,999,948| 139,261
120,205,690,315,959,428203| 100,000,000,000,000,094| 300 1,023693| 30
1,202,056,903,159,594,285057| 1,000,000| 1,000202,000,000,317057| 646,394 65<
12,020,570| 10,000,000| 1,202,057| 141
runtime 0.002 s
120,205,685| 100,000,000| 20,743| 301
</pre>
1,202,056,919| 1,000,000,000| 215,461| 645<
12,020,569,022| 10,000,000,000| 1,322,977| 1,392
120,205,690,298| 100,000,000,000| 145,823| 3,003
1,202,056,903,137| 1,000,000,000,000|400,685,634,379| 6,465<
12,020,569,031,641| 10,000,000,000,000| 1,498,751| 13,924
120,205,690,315,927| 100,000,000,000,000| 57,349| 30,006
1,202,056,903,159,489| 1,000,000,000,000,000| 74,509,198,733| 64,643<
12,020,569,031,596,003| 10,000,000,000,000,000| 0|139,261
120,205,690,315,959,316| 100,000,000,000,000,000| 0|300,023
1,202,056,903,159,593,905| 1,000,000,000,000,000,000| 89,387|646,394<
runtime 0.008 s //best value.
real 0m0,013s
Tested with Apéry´s Constant approximation of 1.000000000000000
runtime 0.065 s
real 0m0,071s</pre>
 
=={{header|Phix}}==
Line 1,063 ⟶ 1,225:
The 10 millionth term is now found in 0.85 seconds and the 1 billionth in about 94 seconds.
 
However, a lot of memory is needed for the sieve since all values in Wren (including bools) need 8 bytes of storage each.
 
We could use only 1/32nd as much memory by importing the BitArray class from [[:Category:Wren-array|Wren-array]] (see program comments for changes needed). However, unfortunately this is much slower to index than a normal List of booleans and the 10 millionth term would now take just over 2 seconds to find and the 1 billionth just under 4 minutes.
<syntaxhighlight lang="wren">import "./math" for Int
//import "./array" for BitArray
import "./fmt" for Fmt
 
var cubeFreeSieve = Fn.new { |n|
var cubeFree = List.filled(n+1, true) // or BitArray.new(n+1, true)
var primes = Int.primeSieve(n.cbrt.ceil)
for (p in primes) {
PureFox
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