Sequence: nth number with exactly n divisors: Difference between revisions

Calculate the sequence where each term <strong>a_n</strong> is the <strong>n^th</strong> that has '''n''' divisors.

 

:*[[Sequence: smallest number greater than previous term with exactly n divisors]]

:*[[Sequence: smallest number with exactly n divisors]]

 

=={{header|Go}}==

 

The remaining terms (up to the 33rd) are not particularly large and so are calculated by brute force.

 

import (

}

}

 

{{out}}

</pre>

 

 

1. Apart from the 2nd and 10th terms, all the even terms are themselves even.

3. While searching for the nth number with exactly n divisors, where feasible a record is kept of any numbers found to have exactly k divisors (k > n) so that the search for these numbers can start from a higher base.

 

 

import (

}

}

 

{{out}}

45 : 327184

</pre>

 

=={{header|Java}}==

import java.math.BigInteger;

 

 

}

}

}

 

int count = 1;

}

}

}

return count;

}

}

}

}

}

 

{{out}}

<pre>

10 : 405

11 : 819628286980801

32 : 3990

33 : 12166144

</pre>

 

=={{header|Kotlin}}==

{{trans|Go}}

 

import java.math.BigInteger

}

}

 

{{output}}

33 : 12166144

</pre>

 

=={{header|Perl}}==

{{libheader|ntheory}}

use warnings;

use bigint;

}

}

{{out}}

<pre>First 20 terms of OEIS:A073916

1 3 25 14 14641 44 24137569 70 1089 405 819628286980801 160 22563490300366186081 2752 9801 462 21559177407076402401757871041 1044 740195513856780056217081017732809 1520</pre>

 

{{works with|Rakudo|2019.03}}

 

[https://tio.run/##dVLbTsJAEH3nKw4IpgW6ATQY2QAao4kvmsijEFPoVpr05nZr2hh@yk/wx@p0CwUf3JfuzJwzc/ZMYyH9cVEk6RqO92ltojRUMJaZia8G6EihUhliBM9FxrzEiqUXCK5rPde3CTwEY1RLPqRirh9F0mSBHU9gzbB09m3Kk4a@SBJk6IDSCHIsc0wppsFwOCiYUmU@p1uzCSPvwzGx0/xdg74NorR9L/AU0UYDXmVutKKEUu9SxNS4VoldH0PGuryxx7xW7BXHGSqE2grYUto5w@Lu6YU6xqlC68GTiTrMUkIGCSIXz/ePi8nt4OriejhucY00qH8FM9k2j4U0JqO1rTbbowftcO@BQbcZLmGHDiVrlSa9WNdFFpcQC8M@asE6XplkiMY40YmhpR0evXvA/6ZIsK0iSRWidzq0PPK0VNo1tbH6Vunr/nwfyTWTueX7JyejyhOKTB2WSI1pWey8/mf4v9BerxRZavmLab/VYb1jXhS/ Try it online!]

 

return 2 if x.is-prime;

+flat (1 .. x.sqrt.floor).map: -> \d {

}

}

 

<pre>First 20 terms of OEIS:A073916

Programming note: &nbsp; this REXX version has minor optimization, and all terms of the sequence are determined (found) in order.

===little optimization===

parse arg N . /*obtain optional argument from the CL.*/

if N=='' | N=="," then N= 15 /*Not specified? Then use the default.*/

w= 50 /*W: width of the 2nd column of output*/

say '─divisors─' center("the Nth number with exactly N divisors", w, '─') /*title.*/

commas: parse arg _; do j=length(_)-3 to 1 by -3; _=insert(',', _, j); end; return _

pPow: numeric digits 1000; return @.i**(i-1) /*temporarily increase decimal digits. */

/*──────────────────────────────────────────────────────────────────────────────────────*/

#divs: procedure; parse arg x 1 y /*X and Y: both set from 1st argument.*/

else do; #= 3; y= x%2; end /*Even? " " " " 3.*/

/* [↑] start with known num of Pdivs.*/

if x//k==0 then do /*if no remainder, then found a divisor*/

#=#+2; y=x%k /*bump # Pdivs, calculate limit Y. */

do j=11 by 6 until j*j>#; if # // j==0 | # // (J+2)==0 then return 0

end /*j*/ /* ___ */

{{out|output|text=&nbsp; when using the input: &nbsp; &nbsp; <tt> 45 </tt>}}

─divisors─ ───────────────────────────────────────────the Nth number with exactly N divisors──────────────────────────────────────────────

1 1

4 14

5 14,641

6 44

7 24,137,569

9 1,089

10 405

11 819,628,286,980,801

12 160

14 2,752

15 9,801

16 462

17 21,559,177,407,076,402,401,757,871,041

19 740,195,513,856,780,056,217,081,017,732,809

20 1,520

21 141,376

22 84,992

24 1,170

25 52,200,625

26 421,888

27 52,900

29 1,116,713,952,456,127,112,240,969,687,448,211,536,647,543,601,817,400,964,721

30 6,768

31 1,300,503,809,464,370,725,741,704,158,412,711,229,899,345,159,119,325,157,292,552,449

32 3,990

34 9,764,864

35 446,265,625

36 5,472

37 11,282,036,144,040,442,334,289,838,466,416,927,162,302,790,252,609,308,623,697,164,994,458,730,076,798,801

39 90,935,296

40 10,416

41 1,300,532,588,674,810,624,476,094,551,095,787,816,112,173,600,565,095,470,117,230,812,218,524,514,342,511,947,837,104,801

42 46,400

::::::* &nbsp; odd non-prime numbers.

::::::* &nbsp; even numbers.

parse arg N . /*obtain optional argument from the CL.*/

if N=='' | N=="," then N= 15 /*Not specified? Then use the default.*/

@.1= 2; Ps= 1; !.= 0; !.1= 2 /*1st prime; number of primes (so far)*/

do p=3 until Ps==N**3 /* [↓] gen N primes, store in @ array.*/

done: do f=N by -1 for N-3; if \zfin.f then return 0; end; return 1

pPow: numeric digits 2000; return @.j**(j-1) /*temporarily increase decimal digits. */

/*──────────────────────────────────────────────────────────────────────────────────────*/

#divs: procedure; parse arg x 1 y /*X and Y: both set from 1st argument.*/

else do; #= 3; y= x%2; end /*Even? " " " " 3.*/

/* [↑] start with known num of Pdivs.*/

if x//k==0 then do /*if no remainder, then found a divisor*/

#=#+2; y=x%k /*bump # Pdivs, calculate limit Y. */

if #//(i+2)==0 then return 0

end /*i*/ /* ___ */

{{out|output|text=&nbsp; is identical to the 1^st REXX version.}} <br><br>

 

=={{header|Sidef}}==

n.prime**(n-1)

}

}

 

{{out}}

<pre>

[1, 3, 25, 14, 14641, 44, 24137569, 70, 1089, 405, 819628286980801, 160, 22563490300366186081, 2752, 9801, 462, 21559177407076402401757871041, 1044, 740195513856780056217081017732809, 1520]

</pre>

 

 

[[Extensible prime generator#zkl]] could be used instead.

p:=BI(1);

primes:=pmax.pump(List(0), p.nextPrime, "copy"); //-->(0,3,5,7,11,13,17,19,...)

}

}

{{out}}

<pre>