Rosetta Code

Wolstenholme numbers

Revision as of 08:10, 3 May 2023 by PureFox (talk | contribs) (Added Wren)

A Wolstenholme number is a number that is the fully reduced numerator of a second order harmonic number:

Wolstenholme numbers is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Prime Wolstenholme numbers are Wolstenholme numbers that are prime.

(Oddly enough, there is also a class of numbers called Wolstenholme primes that are not Wolstenholme numbers that are prime. This is not them.)


Task
  • Find and display the first 20 Wolstenholme numbers. (Or as many as reasonably supported by your language if it is fewer.)
  • Find and display the first 4 prime Wolstenholme numbers. (Or as many as reasonably supported by your language if it is fewer.)


Stretch
  • Find and display the digit count of the 500th, 1000th, 2500th, 5000th, 10000th Wolstenholme numbers.
  • Find and display the digit count of the first 15 prime Wolstenholme numbers.


See also


J

NB. the first y members of the Wholstenhome sequence
WolstenholmeSeq=: {{ 2 {."1@x:+/\%*:1x+i.y }}

   WolstenholmeSeq 20
1 5 49 205 5269 5369 266681 1077749 9778141 1968329 239437889 240505109 40799043101 40931552621 205234915681 822968714749 238357395880861 238820721143261 86364397717734821 17299975731542641
   (#~ 1 p: ])WolstenholmeSeq 20
5 266681 40799043101 86364397717734821

Python

""" rosettacode.orgwiki/Magnanimous_numbers """

from fractions import Fraction
from sympy import isprime


def wolstenholme(k):
    """ Wolstenholme numbers """
    return sum(Fraction(1, i*i) for i in range(1, k+1)).numerator

def process_wolstenholmes(nmax):
    """ Run the tasks at rosettacode.org/wiki/Wolstenholme_numbers """
    wols = [wolstenholme(n) for n in range(1, nmax+1)]
    print('First 20 Wolstenholme numbers:')
    for i in wols[:20]:
        print(i)
    print('\nFour Wolstenholme primes:')
    for j in filter(isprime, wols):
        print(j)


process_wolstenholmes(20)
Output:
First 20 Wolstenholme numbers:
1
5
49
205
5269
5369
266681
1077749
9778141
1968329
239437889
240505109
40799043101
40931552621
205234915681
822968714749
238357395880861
238820721143261
86364397717734821
17299975731542641

Four Wolstenholme primes:
5
266681
40799043101
86364397717734821

Raku

use Lingua::EN::Numbers;

sub abbr ($_) { .chars < 41 ?? $_ !! .substr(0,20) ~ '..' ~ .substr(*-20) ~ " (digits: {.chars})" }

my @wolstenholme = lazy ([\+] (1..∞).hyper.map: {FatRat.new: 1, .²}).map: *.numerator;

say 'Wolstenholme numbers:';
printf "%8s: %s\n", .&ordinal-digit(:c), abbr @wolstenholme[$_-1] for flat 1..20, 5e2, 1e3, 2.5e3, 5e3, 1e4;

say "\nPrime Wolstenholme numbers:";
printf "%8s: %s\n", .&ordinal-digit(:c), @wolstenholme.grep(&is-prime)[$_-1]».&abbr for 1..15;
Output:
Wolstenholme numbers:
     1st: 1
     2nd: 5
     3rd: 49
     4th: 205
     5th: 5269
     6th: 5369
     7th: 266681
     8th: 1077749
     9th: 9778141
    10th: 1968329
    11th: 239437889
    12th: 240505109
    13th: 40799043101
    14th: 40931552621
    15th: 205234915681
    16th: 822968714749
    17th: 238357395880861
    18th: 238820721143261
    19th: 86364397717734821
    20th: 17299975731542641
   500th: 40989667509417020364..48084984597965892703 (digits: 434)
 1,000th: 83545938483149689478..99094240207812766449 (digits: 866)
 2,500th: 64537911900230612090..12785535902976933153 (digits: 2164)
 5,000th: 34472086597488537716..22525144829082590451 (digits: 4340)
10,000th: 54714423173933343999..49175649667700005717 (digits: 8693)

Prime Wolstenholme numbers:
     1st: 5
     2nd: 266681
     3rd: 40799043101
     4th: 86364397717734821
     5th: 36190908596780862323..79995976006474252029 (digits: 104)
     6th: 33427988094524601237..48446489305085140033 (digits: 156)
     7th: 22812704758392002353..84405125167217413149 (digits: 218)
     8th: 28347687473208792918..45794572911130248059 (digits: 318)
     9th: 78440559440644426017..30422337523878698419 (digits: 520)
    10th: 22706893975121925531..02173859396183964989 (digits: 649)
    11th: 27310394808585898968..86311385662644388271 (digits: 935)
    12th: 13001072736642048751..08635832246554146071 (digits: 984)
    13th: 15086863305391456002..05367804007944918693 (digits: 1202)
    14th: 23541935187269979100..02324742766220468879 (digits: 1518)
    15th: 40306783143871607599..58901192511859288941 (digits: 1539)

Wren

Library: Wren-gmp
Library: Wren-fmt
import "./gmp" for Mpq
import "./fmt" for Fmt

var w = Mpq.new()
var h = Mpq.new()
var primes = []
var l = [500, 1000, 2500, 5000, 10000]
System.print("Wolstenholme numbers:")
for (k in 1..10000) {
    h.set(1, k * k)
    w.add(h)
    var n = w.num
    if (primes.count < 15 && n.probPrime(15) > 0) primes.add(n)
    if (k <= 20) {
        Fmt.print("$8r: $i", k, n)
    } else if (l.contains(k)) {
        var ns = n.toString
        Fmt.print("$,8r: $20a (digits: $d)", k, ns, ns.count)      
    }
}
System.print("\nPrime Wolstenholme numbers:")
for (i in 0...primes.count) {
    if (i < 4) {
        Fmt.print("$8r: $i", i+1, primes[i])
    } else {
        var ps = primes[i].toString
        Fmt.print("$8r: $20a (digits: $d)", i+1, ps, ps.count)
    }
}
Output:
Wolstenholme numbers:
     1st: 1
     2nd: 5
     3rd: 49
     4th: 205
     5th: 5269
     6th: 5369
     7th: 266681
     8th: 1077749
     9th: 9778141
    10th: 1968329
    11th: 239437889
    12th: 240505109
    13th: 40799043101
    14th: 40931552621
    15th: 205234915681
    16th: 822968714749
    17th: 238357395880861
    18th: 238820721143261
    19th: 86364397717734821
    20th: 17299975731542641
   500th: 40989667509417020364...48084984597965892703 (digits: 434)
 1,000th: 83545938483149689478...99094240207812766449 (digits: 866)
 2,500th: 64537911900230612090...12785535902976933153 (digits: 2164)
 5,000th: 34472086597488537716...22525144829082590451 (digits: 4340)
10,000th: 54714423173933343999...49175649667700005717 (digits: 8693)

Prime Wolstenholme numbers:
     1st: 5
     2nd: 266681
     3rd: 40799043101
     4th: 86364397717734821
     5th: 36190908596780862323...79995976006474252029 (digits: 104)
     6th: 33427988094524601237...48446489305085140033 (digits: 156)
     7th: 22812704758392002353...84405125167217413149 (digits: 218)
     8th: 28347687473208792918...45794572911130248059 (digits: 318)
     9th: 78440559440644426017...30422337523878698419 (digits: 520)
    10th: 22706893975121925531...02173859396183964989 (digits: 649)
    11th: 27310394808585898968...86311385662644388271 (digits: 935)
    12th: 13001072736642048751...08635832246554146071 (digits: 984)
    13th: 15086863305391456002...05367804007944918693 (digits: 1202)
    14th: 23541935187269979100...02324742766220468879 (digits: 1518)
    15th: 40306783143871607599...58901192511859288941 (digits: 1539)
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