Harmonic series: Difference between revisions
ReeceGoding (talk | contribs) Added R. |
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9 : 4550
10 : 12367</pre>
=={{header|R}}==
The talk page helpfully points out that we can be remarkably lazy here.
<lang R>HofN <- function(n) sum(1/seq_len(n)) #Task 1
H <- sapply(1:100000, HofN)
print(H[1:20]) #Task 2
print(sapply(1:10, function(x) which.max(H > x))) #Task 3 and stretch</lang>
{{out}}
<pre>> print(H[1:20]) #Task 2
[1] 1.000000 1.500000 1.833333 2.083333 2.283333 2.450000 2.592857 2.717857 2.828968 2.928968 3.019877 3.103211 3.180134 3.251562
[15] 3.318229 3.380729 3.439553 3.495108 3.547740 3.597740
> print(sapply(1:10, function(x) which.max(H > x))) #Task 3 and stretch
[1] 2 4 11 31 83 227 616 1674 4550 12367</pre>
=={{header|Raku}}==
|
Revision as of 15:40, 10 October 2021
This page uses content from Wikipedia. The original article was at Harmonic number. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance) |
In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers:
Hn = 1 + 1/2 + 1/3 + ... + 1/n
The series of harmonic numbers thus obtained is often loosely referred to as the harmonic series.
Harmonic numbers are closely related to the Riemann zeta function, and roughly approximate the natural logarithm function; differing by γ (lowercase Gamma), the Euler–Mascheroni constant.
The harmonic series is divergent, albeit quite slowly, and grows toward infinity.
- Task
- Write a function (routine, procedure, whatever it may be called in your language) to generate harmonic numbers.
- Use that procedure to show the values of the first 20 harmonic numbers.
- Find and show the position in the series of the first value greater than the integers 1 through 5
- Stretch
- Find and show the position in the series of the first value greater than the integers 6 through 10
- Related
C++
<lang cpp>#include <iomanip>
- include <iostream>
- include <boost/rational.hpp>
- include <boost/multiprecision/gmp.hpp>
using integer = boost::multiprecision::mpz_int; using rational = boost::rational<integer>;
class harmonic_generator { public:
rational next() { rational result = term_; term_ += rational(1, ++n_); return result; } void reset() { n_ = 1; term_ = 1; }
private:
integer n_ = 1; rational term_ = 1;
};
int main() {
std::cout << "First 20 harmonic numbers:\n"; harmonic_generator hgen; for (int i = 1; i <= 20; ++i) std::cout << std::setw(2) << i << ". " << hgen.next() << '\n'; rational h; for (int i = 1; i <= 80; ++i) h = hgen.next(); std::cout << "\n100th harmonic number: " << h << "\n\n";
int n = 1; hgen.reset(); for (int i = 1; n <= 10; ++i) { if (hgen.next() > n) std::cout << "Position of first term > " << std::setw(2) << n++ << ": " << i << '\n'; }
}</lang>
- Output:
First 20 harmonic numbers: 1. 1/1 2. 3/2 3. 11/6 4. 25/12 5. 137/60 6. 49/20 7. 363/140 8. 761/280 9. 7129/2520 10. 7381/2520 11. 83711/27720 12. 86021/27720 13. 1145993/360360 14. 1171733/360360 15. 1195757/360360 16. 2436559/720720 17. 42142223/12252240 18. 14274301/4084080 19. 275295799/77597520 20. 55835135/15519504 100th harmonic number: 14466636279520351160221518043104131447711/2788815009188499086581352357412492142272 Position of first term > 1: 2 Position of first term > 2: 4 Position of first term > 3: 11 Position of first term > 4: 31 Position of first term > 5: 83 Position of first term > 6: 227 Position of first term > 7: 616 Position of first term > 8: 1674 Position of first term > 9: 4550 Position of first term > 10: 12367
Factor
This solution uses the following (rather accurate) approximation of the harmonic numbers to find the first indices greater than the integers:
Hn ≈ ln(n) + γ + 1/2n - 1/12n2
where γ
is the Euler-Mascheroni constant, approximately 0.5772156649
.
<lang factor>USING: formatting grouping io kernel lists lists.lazy math math.functions math.ranges math.statistics math.text.english prettyprint sequences tools.memory.private ;
! Euler-Mascheroni constant CONSTANT: γ 0.5772156649
- Hn-approx ( n -- ~Hn )
[ log γ + 1 2 ] [ * /f + 1 ] [ sq 12 * /f - ] tri ;
- lharmonics ( -- list ) 1 lfrom [ Hn-approx ] lmap-lazy ;
- first-gt ( m -- n ) lharmonics swap '[ _ < ] lwhile llength ;
"First twenty harmonic numbers as mixed numbers:" print 100 [1,b] [ recip ] map cum-sum [ 20 head 5 group simple-table. nl ] [ "One hundredth:" print last . nl ] bi
"(zero based) Index of first value:" print 10 [1,b] [
dup first-gt [ commas ] [ 1 + number>text ] bi " greater than %2d: %6s (term number %s)\n" printf
] each</lang>
- Output:
First twenty harmonic numbers as mixed numbers: 1 1+1/2 1+5/6 2+1/12 2+17/60 2+9/20 2+83/140 2+201/280 2+2089/2520 2+2341/2520 3+551/27720 3+2861/27720 3+64913/360360 3+90653/360360 3+114677/360360 3+274399/720720 3+5385503/12252240 3+2022061/4084080 3+42503239/77597520 3+9276623/15519504 One hundredth: 5+522561233577855727314756256041670736351/2788815009188499086581352357412492142272 (zero based) Index of first value: greater than 1: 1 (term number two) greater than 2: 3 (term number four) greater than 3: 10 (term number eleven) greater than 4: 30 (term number thirty-one) greater than 5: 82 (term number eighty-three) greater than 6: 226 (term number two hundred and twenty-seven) greater than 7: 615 (term number six hundred and sixteen) greater than 8: 1,673 (term number one thousand, six hundred and seventy-four) greater than 9: 4,549 (term number four thousand, five hundred and fifty) greater than 10: 12,366 (term number twelve thousand, three hundred and sixty-seven)
Go
<lang go>package main
import (
"fmt" "math/big"
)
func harmonic(n int) *big.Rat {
sum := new(big.Rat) for i := int64(1); i <= int64(n); i++ { r := big.NewRat(1, i) sum.Add(sum, r) } return sum
}
func main() {
fmt.Println("The first 20 harmonic numbers and the 100th, expressed in rational form, are:") numbers := make([]int, 21) for i := 1; i <= 20; i++ { numbers[i-1] = i } numbers[20] = 100 for _, i := range numbers { fmt.Printf("%3d : %s\n", i, harmonic(i)) }
fmt.Println("\nThe first harmonic number to exceed the following integers is:") const limit = 10 for i, n, h := 1, 1, 0.0; i <= limit; n++ { h += 1.0 / float64(n) if h > float64(i) { fmt.Printf("integer = %2d -> n = %6d -> harmonic number = %9.6f (to 6dp)\n", i, n, h) i++ } }
}</lang>
- Output:
The first 20 harmonic numbers and the 100th, expressed in rational form, are: 1 : 1/1 2 : 3/2 3 : 11/6 4 : 25/12 5 : 137/60 6 : 49/20 7 : 363/140 8 : 761/280 9 : 7129/2520 10 : 7381/2520 11 : 83711/27720 12 : 86021/27720 13 : 1145993/360360 14 : 1171733/360360 15 : 1195757/360360 16 : 2436559/720720 17 : 42142223/12252240 18 : 14274301/4084080 19 : 275295799/77597520 20 : 55835135/15519504 100 : 14466636279520351160221518043104131447711/2788815009188499086581352357412492142272 The first harmonic number to exceed the following integers is: integer = 1 -> n = 2 -> harmonic number = 1.500000 (to 6dp) integer = 2 -> n = 4 -> harmonic number = 2.083333 (to 6dp) integer = 3 -> n = 11 -> harmonic number = 3.019877 (to 6dp) integer = 4 -> n = 31 -> harmonic number = 4.027245 (to 6dp) integer = 5 -> n = 83 -> harmonic number = 5.002068 (to 6dp) integer = 6 -> n = 227 -> harmonic number = 6.004367 (to 6dp) integer = 7 -> n = 616 -> harmonic number = 7.001274 (to 6dp) integer = 8 -> n = 1674 -> harmonic number = 8.000486 (to 6dp) integer = 9 -> n = 4550 -> harmonic number = 9.000208 (to 6dp) integer = 10 -> n = 12367 -> harmonic number = 10.000043 (to 6dp)
Haskell
<lang haskell>import Data.List (find) import Data.Ratio
HARMONIC SERIES --------------------
harmonic :: [Rational] harmonic =
scanl1 (\a x -> a + 1 / x) [1 ..]
TESTS -------------------------
main :: IO () main = do
putStrLn "First 20 terms:" mapM_ putStrLn $ showRatio <$> take 20 harmonic
putStrLn "\n100th term:" putStrLn $ showRatio (harmonic !! 99) putStrLn ""
putStrLn "One-based indices of first terms above threshold values:" let indexedHarmonic = zip [0 ..] harmonic mapM_ putStrLn $ fmap ( showFirstLimit <*> \n -> find ((> n) . snd) indexedHarmonic ) [1 .. 10]
DISPLAY FORMATTING ------------------
showFirstLimit n (Just (i, r)) =
"Term " <> show (succ i) <> " is the first above " <> show (numerator n)
showRatio :: Ratio Integer -> String showRatio =
((<>) . show . numerator) <*> (('/' :) . show . denominator)</lang>
- Output:
First 20 terms: 1/1 3/2 11/6 25/12 137/60 49/20 363/140 761/280 7129/2520 7381/2520 83711/27720 86021/27720 1145993/360360 1171733/360360 1195757/360360 2436559/720720 42142223/12252240 14274301/4084080 275295799/77597520 55835135/15519504 100th term: 14466636279520351160221518043104131447711/2788815009188499086581352357412492142272 One-based indices of first terms above threshold values: Term 2 is the first above 1 Term 4 is the first above 2 Term 11 is the first above 3 Term 31 is the first above 4 Term 83 is the first above 5 Term 227 is the first above 6 Term 616 is the first above 7 Term 1674 is the first above 8 Term 4550 is the first above 9 Term 12367 is the first above 10
Julia
<lang julia>const memoizer = [BigFloat(1.0), BigFloat(1.5)]
"""
harmonic(n::Integer)::BigFloat
Calculates harmonic numbers. The integer argument `n` should be positive. """ function harmonic(n::Integer)::BigFloat
if n < 0 throw(DomainError(n)) elseif n == 0 return BigFloat(0.0) # by convention elseif length(memoizer) >= n return memoizer[n] elseif length(memoizer) + 1 == n h = memoizer[end] + BigFloat(1.0) / n push!(memoizer, h) return h elseif n < 1_000_000 start, x = length(memoizer), memoizer[end] for i in start+1:n push!(memoizer, (x += big"1.0" / i)) end return memoizer[end] else # use H(n) = eulergamma + digamma(n + 1), instead, if memory use of memoization too large x = n + big"1.0" digam = BigFloat() ccall((:mpfr_digamma, :libmpfr), Int32, (Ref{BigFloat}, Ref{BigFloat}, Int32), digam, x, 1) return Base.MathConstants.eulergamma + digam end
end
function testharmonics(upperlimit = 11)
n = 1 while (h = harmonic(n)) < upperlimit nextintegerfloor = h < 1.8 ? h > 1.0 : floor(h) > floor(memoizer[n - 1]) if n < 21 || nextintegerfloor println("harmonic($n) = $h") nextintegerfloor && println(" $n is also the term number for the first harmonic > $(floor(h))") end n += 1 end
end
testharmonics()
</lang>
- Output:
harmonic(1) = 1.0 harmonic(2) = 1.5 2 is also the term number for the first harmonic > 1.0 harmonic(3) = 1.833333333333333333333333333333333333333333333333333333333333333333333333333339 harmonic(4) = 2.083333333333333333333333333333333333333333333333333333333333333333333333333356 4 is also the term number for the first harmonic > 2.0 harmonic(5) = 2.283333333333333333333333333333333333333333333333333333333333333333333333333363 harmonic(6) = 2.450000000000000000000000000000000000000000000000000000000000000000000000000041 harmonic(7) = 2.592857142857142857142857142857142857142857142857142857142857142857142857142913 harmonic(8) = 2.717857142857142857142857142857142857142857142857142857142857142857142857142913 harmonic(9) = 2.828968253968253968253968253968253968253968253968253968253968253968253968254009 harmonic(10) = 2.928968253968253968253968253968253968253968253968253968253968253968253968253995 harmonic(11) = 3.019877344877344877344877344877344877344877344877344877344877344877344877344889 11 is also the term number for the first harmonic > 3.0 harmonic(12) = 3.103210678210678210678210678210678210678210678210678210678210678210678210678211 harmonic(13) = 3.180133755133755133755133755133755133755133755133755133755133755133755133755123 harmonic(14) = 3.251562326562326562326562326562326562326562326562326562326562326562326562326542 harmonic(15) = 3.318228993228993228993228993228993228993228993228993228993228993228993228993199 harmonic(16) = 3.380728993228993228993228993228993228993228993228993228993228993228993228993199 harmonic(17) = 3.439552522640757934875581934405463817228523110876052052522640757934875581934384 harmonic(18) = 3.495108078196313490431137489961019372784078666431607608078196313490431137489932 harmonic(19) = 3.547739657143681911483769068908387793836710245378976029130827892437799558542556 harmonic(20) = 3.597739657143681911483769068908387793836710245378976029130827892437799558542549 harmonic(31) = 4.027245195436520102759838180253409570739320924649712368107240380481568735938418 31 is also the term number for the first harmonic > 4.0 harmonic(83) = 5.002068272680166053728324750753870264345455215566438587478989543061001039767003 83 is also the term number for the first harmonic > 5.0 harmonic(227) = 6.004366708345566023376436217157408474650893771305512336984772241757969069086895 227 is also the term number for the first harmonic > 6.0 harmonic(616) = 7.001274097134160381487068933022945074864048309674852535721112060499845844673362 616 is also the term number for the first harmonic > 7.0 harmonic(1674) = 8.000485571995779067790304796519697445800341927883408389172647367923220595045883 1674 is also the term number for the first harmonic > 8.0 harmonic(4550) = 9.000208062931140339164179501268928624268799275400095995932594006439583360896694 4550 is also the term number for the first harmonic > 9.0 harmonic(12367) = 10.00004300827580769470675707492981720768686887243344211163998834649135547210551 12367 is also the term number for the first harmonic > 10.0
Using rationals
<lang julia>const harmonics = accumulate((x, y) -> x + big"1" // y, 1:12370)
println("First twenty harmonic numbers as rationals:") foreach(i -> println(rpad(i, 3), " => ", harmonics[i]), 1:20)
println("\nThe 100th harmonic is: ", harmonics[100], "\n")
for n in 1:10
idx = findfirst(x -> x > n, harmonics) print("First Harmonic > $n is at position $idx and is: ", harmonics[idx], "\n\n")
end
</lang>
- Output:
First twenty harmonic numbers as rationals: 1 => 1//1 2 => 3//2 3 => 11//6 4 => 25//12 5 => 137//60 6 => 49//20 7 => 363//140 8 => 761//280 9 => 7129//2520 10 => 7381//2520 11 => 83711//27720 12 => 86021//27720 13 => 1145993//360360 14 => 1171733//360360 15 => 1195757//360360 16 => 2436559//720720 17 => 42142223//12252240 18 => 14274301//4084080 19 => 275295799//77597520 20 => 55835135//15519504 The 100th harmonic is: 14466636279520351160221518043104131447711//2788815009188499086581352357412492142272 First Harmonic > 1 is at position 2 and is: 3//2 First Harmonic > 2 is at position 4 and is: 25//12 First Harmonic > 3 is at position 11 and is: 83711//27720 First Harmonic > 4 is at position 31 and is: 290774257297357//72201776446800 First Harmonic > 5 is at position 83 and is: 3672441655127796364812512959533039359//734184632222154704090370027645633600 First Harmonic > 6 is at position 227 and is: 7210530454341478178114292924106791866448071719960766673184657267908514585008387695857601640547547//1200881092808579751109445892858157237623011602251376919557525378451885327053551694768211209584000 First Harmonic > 7 is at position 616 and is: 32418148234584438506244669620559801586453637524407441287287956498408167324384836135154335608450103705554364675635095796463552565723833053990601655182066284053456674137000162471237344506126617371216079120095971345235097691744446098739930413505848076716148051590567//4630321250792651667959778962730272725135972962879403462498154236164728708558353376685441262094879300811505205446228182535680317471673227192574715015221837802633281044802046825192844588381756912000006314261374254853934981337617963082619967597058012665881508170240 First Harmonic > 8 is at position 1674 and is: 138141199730356031917751723608451630140926572041533266161867229808423134343694448776284861362361049658120307218628922472967074569643922410693236711646552594017799177544898923160715275467074773052672543691485042344006328346792717577331346270889819765648723902241194752204176829233330048155240810379245143159803746553605910654524991868894244655125224374793405027930983653080599361646248333997230466686371943554425849838679719632376766617639339476485319908534320267241696294676687857612211224929536683459738832437894009105247288571220395942120501422698013842452465990861198601652372525840473807472136998877762025722745653584203415416989419985647929410780044991971375219261681255528981314607252039408935696726675566989805504130221402879//17266601943998720215934555231097205020805360283925021776131771421603665580550992851730546192114435876915764081799263950036774496079790573533431946151195888729825227967898856935376506079740151630837994958679444177020560516804785312006925540853535681555388532126434276232761609235943363516882249874992501288848101663246418034482830782765799910587627015490435618612817485429045075697874945013812237906815356826216080083918931352849328226757321069450237292900402036556169613799106635542806391854137317272454464723030848621364955770185788658456980359320481573473968897297574022422610618929844739054324826111937140267616883649616842817036401014150739236179076410461198210703989272527503945999049527912447331310263532711648780174245760000 First Harmonic > 9 is at position 4550 and is: 2803922057204811543989535496612889221797101689401339733425779071893883726062730194418722759045899108563649222513327676111056373937430194191361767957531399947657882950860187200723323262653682200418079207248717851019834955117510807290247517344853824182604853734193915236890382944826941254750728989773414206859570065960640186792890664839103798433780558510849879082382309565191620282894846917562664482131202291813204472491830923673050493632880376511556753354676402403084278296415856939754434734260623967677761451430096042117604187063669249143393076890252523001247824132437003474131230135624887874799541971404156392012052221712958314528322355541629626096561834856317166367946519841625273109689443740339231545297896759513984852874078072318627511091395704617775835151474534768408675032414448963002972590468720049363107497714657975180640114587539853156735035634359134035816625004885128358508857789053399591546078553132392557669776329295754833648165907192091248381853554990310275294735796601315160268572473977239894244510958924644461475601889585298550864600912608515846981406115574447937712467306028124552530287697424685820096894634400504911945644842525080877704081133701601474836419212357427281193595545654309059483619700606698188346807113314394305087007324692154551457096864441413765832492015835459544797423019196304195496229963766867559041682555605485756372169731071238590755894190062578370776223859945908951232125318924762539326986977852885933136208484015119262988152215227210365689996539057011922584498047033647706625149283085817504520106280560129638082705378996661011957957116006944584486439354147266379272818775057457727700062964206158298239058238442033871955799108732891594547873095023995742714560400884479305635074935996707008596929664779547461570776674459690821497530879010144224813067523793975748941040890361283562073364309054362072137372458221297707667641071756676535258762295067395479041974831597965979613081474695278123265026663091006352059826017007472704096220374679625403//311539693038123496722215310551186904103082524726970170658802584453586817702845967012058559851648961565162714074357248615113237727615741409397613403942762027416418250197869235139807433555974792925305500675331577565501246819843747067123329275977858482555209853394601249731116273203314723492820718920663543363350452108859588055589207296367576316485295250028686463172112871115738305282269830733309445640796833467195761172540352742868324044527033042113461149657815168691435568328007659916683562048346389048741726140068665849102873300936534794943981012153540913055621237325128266901087332680399450826342551172911601227269655973249675723840414635853583858640961466968497444812002669481333031399627561658239662030094505016308687209615676717899935366285911202891060019766201048122485091665890488010192945266722188069264271709669076333959826410108104815827793429989042734394907438794428685359256959974813630940895555145059961227687918900885009633119015801758204737016204228694445315173976166116707848897795690334687813067638452214980721143416739719469158748498888549650173439285892765248219730073273036517725939409971809083058041780264560576764115967213013380206017646943405476909576091357482868476761698005324735669002986992261514164471647621973497050651105016702049275202360233254308086080840811811270395182950866854592166254651076234420662875446908943737749367221987863265181700223872816922018831416550743452655599236620182838774266078307310527709836462300005876748274266928980195184478251265986886488170211071786897470230189921260226806813948222307759474759342997757077589856537503856731410785456607398210129322637205537554981139121822428297380197652187006595903988741596680008103907700803965881407891948670316751093348441767470344198247057046756517630261990510817072323901911147038583224901683429072731095666462995891134319532658460042039832285713339493699639220237857309737109028479215274870712938102667472652582674547746273957596211401299308669708874527010828161490844069542400000 First Harmonic > 10 is at position 12367 and is: 45345034307070335134555666635024887773984040811638642621462405132584886144532377986073358145244201027037357938621772142209616405161627868105778015683798484008416515631723931456447477569347791478527857610769841230294963993979332743283338510233568324905959949579062456226347439170271217301636734446205747735713945863253433483740897523300848534809108461545232146005948850103219707387931993351213767427376889962909120228840233148459893174855648971003785113710072840958355449134583864883502737527932996232809536817702108606704972939587420565678604933039194329667673092895550055241161559979675050092624926868871458018343441582176596250529929191074291128267400687630170072609517611336780875630097770219247592957358603597479920082804823686786983637602905120758432634792571445674234736536086607803064981671382406890757026618258218813838974120950628944981905245622673666105003761865085749235468633696285737487359622886699738348673700526205671244526918181580594955298707753497194308839658138978679642619068841057090153286541341850948526490774435583437961002123092419531093235665933120984675385509883201479274589825686372618722516148263128067561905455373547546391052207792967353055080744047776986080724800254366264659407835575877760954125271799727336398809285045211625324747901193314444326911865397319722296088570348383225778250731285584101470972417783253384871308701259088227660489254359610314686851359252857300051920723671060874215404857586898535480326782181158608408279619195590515799061476749269801594625879703128842040583394154984407245336678544432608949614158673227077737666441262935879700412919345767447983556401804802974663896313057250911344831472616803645836814569048455850429162273597559955668906516180684124175673531692230282389747164102840354665242183576801399969588541389263384471603494896180265191824808290512910947744283695895248414399524148990184306294157590456646295768057794302761262994594722655055964096754993330351814041078876315865417737202330307734602330970139055634545867680529440237996308032770048686378187392782129596233133189016852266246661061281495413831799689680798229084372131339265032436345645554714807426037605589639494475684095551532755566945308404954024645746798191900979340826809913903596772706839217198094879854114314539133185622222803259348108969761096223170124626733975997197050299930954519458027451118459941271137375313026666466736293093998795637218208416937639900402972180842804696606039337369141035805923341690099055369556837577997854655210298104810520933189746485979124018196909319459534410139406110909222206154476323155751314166239864898017483972228615274429236864538159277543058839398436445287447883114413176678000824006252470021316902405765477155539855660847221656031691805464444896245205294585896596429456743028035184934461863654010366528689573439474363008455160522485449967670027033722278013317043478614293767140874291908415829941007066088704322634788409768471256565985280476650542596002212877354416976349052617941581234194408366031879100064003856581616095600774809166667678552494014963764538137429108976853046746079327814801533505923050861435172931240483219080490333386411655709499209676409734678148401497706097394309678808363917083429809690764137065474753611515665686465587299537880762699588786806388064113473982700024684504900602299235464951536685894424056502683548413627701664149406689523427968563342971845946890142203694025409999036915672932674775076177305732160054942888943862654196099307096880517864598688559999274834885952923776254724920526071660112996620623947730034924455037553195043344076284277868014382989931974908029499035370171767959005233662215283043876585561861314525586427883264270058319450472084164619032769031551576680710841970852322393037539325119666017996420868400106357872834174841151636016380407477204298535548979650626662451521208388882826961445471585692683081515085943594719122539526092112699608963277450836088209198655644116097290500948029821534837319698118213632136665010898032500699148625910693981318114217177275819944652811575951322137901737402106063154587879423128895000245926444726344972103433121968004238984107852115590751270399146488824187698443051260706281584394068239758982021239369875085477123234398148089271884961390297847264240658363997995669574653293259007333471580117358907760396222393711792653221029880957923342959879228299043166437440244751583666970116390226900111585349332814785514272218273380355024057390176759649953398455756889742471454303755506911727925831506750404598474949928168799378739227974329175403082516852308081709012535982487502050229124471983831128472775878806304759862969459429435276021291396028499943188538938799118003305779761837824757679798705125552062627109499973531037724930204716526740123383060533255126341151138114409083882474220727550189504138353942985034294716389262234667241206934164898746082763605571675460470098269794908360324625747888989968089590921522328937339714693262302234111258577347278529070921677616792503989869186375043102196295624700036419726681106495130783471156636071196116110061360911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Mathematica /Wolfram Language
<lang Mathematica>nums = HarmonicNumber[Range[15000]]; nums;; 20 LengthWhile[nums, LessEqualThan[#]] + 1 & /@ Range[10]</lang>
- Output:
{1, 3/2, 11/6, 25/12, 137/60, 49/20, 363/140, 761/280, 7129/2520, 7381/2520, 83711/27720, 86021/27720, 1145993/360360, 1171733/360360, 1195757/360360, 2436559/720720, 42142223/12252240, 14274301/4084080, 275295799/77597520, 55835135/15519504} {2, 4, 11, 31, 83, 227, 616, 1674, 4550, 12367}
Nim
Using floats
<lang Nim>import strformat
iterator h(): (int, float) =
## Yield the index of the term and its value. var n = 1 var r = 0.0 while true: r += 1 / n yield (n, r) inc n
echo "First 20 terms of the harmonic series:" for (idx, val) in h():
echo &"{idx:2}: {val}" if idx == 20: break
echo()
var target = 1.0 for (idx, val) in h():
if val > target: echo &"Index of the first term greater than {target.int:2}: {idx}" if target == 10: break else: target += 1</lang>
- Output:
1: 1.0 2: 1.5 3: 1.833333333333333 4: 2.083333333333333 5: 2.283333333333333 6: 2.45 7: 2.592857142857143 8: 2.717857142857143 9: 2.828968253968254 10: 2.928968253968254 11: 3.019877344877345 12: 3.103210678210678 13: 3.180133755133755 14: 3.251562326562327 15: 3.318228993228994 16: 3.380728993228994 17: 3.439552522640758 18: 3.495108078196314 19: 3.547739657143682 20: 3.597739657143682 Index of the first term greater than 1: 2 Index of the first term greater than 2: 4 Index of the first term greater than 3: 11 Index of the first term greater than 4: 31 Index of the first term greater than 5: 83 Index of the first term greater than 6: 227 Index of the first term greater than 7: 616 Index of the first term greater than 8: 1674 Index of the first term greater than 9: 4550 Index of the first term greater than 10: 12367
Using big integers
<lang Nim>import strformat import bignum
iterator h(): (int, Rat) =
var n = 1 var r = newRat() while true: r += newRat(1, n) yield (n, r) inc n
echo "First 20 terms of the harmonic series:" for (idx, val) in h():
echo &"{idx:2}: {val}" if idx == 20: break
echo()
var target = 1 for (idx, val) in h():
if val > target: echo &"Index of the first term greater than {target:2}: {idx}" if target == 10: break else: inc target</lang>
- Output:
1: 1 2: 3/2 3: 11/6 4: 25/12 5: 137/60 6: 49/20 7: 363/140 8: 761/280 9: 7129/2520 10: 7381/2520 11: 83711/27720 12: 86021/27720 13: 1145993/360360 14: 1171733/360360 15: 1195757/360360 16: 2436559/720720 17: 42142223/12252240 18: 14274301/4084080 19: 275295799/77597520 20: 55835135/15519504 Index of the first term greater than 1: 2 Index of the first term greater than 2: 4 Index of the first term greater than 3: 11 Index of the first term greater than 4: 31 Index of the first term greater than 5: 83 Index of the first term greater than 6: 227 Index of the first term greater than 7: 616 Index of the first term greater than 8: 1674 Index of the first term greater than 9: 4550 Index of the first term greater than 10: 12367
Perl
<lang perl>use strict; use warnings; use feature 'say'; use Math::AnyNum ':overload'; use List::AllUtils 'firstidx';
my(@H,$n) = 0; do { ++$n and push @H, $H[-1] + 1/$n } until $H[-1] >= 10; shift @H;
say 'First twenty harmonic numbers as rationals:'; my $c = 0; printf("%20s", $_) and (not ++$c%5) and print "\n" for @H[0..19];
say "\nIndex of first value (zero based):"; for my $i (1..10) {
printf " greater than %2d: %5s\n", $i, firstidx { $_ > $i } @H;
}</lang>
- Output:
First twenty harmonic numbers as rationals: 1 3/2 11/6 25/12 137/60 49/20 363/140 761/280 7129/2520 7381/2520 83711/27720 86021/27720 1145993/360360 1171733/360360 1195757/360360 2436559/720720 42142223/12252240 14274301/4084080 275295799/77597520 55835135/15519504 Index of first value (zero based): greater than 1: 1 greater than 2: 3 greater than 3: 10 greater than 4: 30 greater than 5: 82 greater than 6: 226 greater than 7: 615 greater than 8: 1673 greater than 9: 4549 greater than 10: 12366
Phix
requires("0.8.4") include mpfr.e integer n = 1, gn = 1, lim = iff(platform()=JS?8:10) mpq hn = mpq_init_set_si(1) sequence gt = {} puts(1,"First twenty harmonic numbers as rationals:\n") while gn<=lim do if n<=20 then printf(1,"%18s%s",{mpq_get_str(hn),iff(mod(n,5)?" ","\n")}) end if if n=100 then printf(1,"\nOne Hundredth:\n%s\n\n",{mpq_get_str(hn)}) end if if mpq_cmp_si(hn,gn)>0 then gt &= n gn += 1 end if n += 1 mpq_add_si(hn,hn,1,n) end while printf(1,"(one based) Index of first value:\n") for i=1 to length(gt) do printf(1," greater than %2d: %,6d (%s term)\n",{i,gt[i],ordinal(gt[i])}) end for
- Output:
First twenty harmonic numbers as rationals: 1 3/2 11/6 25/12 137/60 49/20 363/140 761/280 7129/2520 7381/2520 83711/27720 86021/27720 1145993/360360 1171733/360360 1195757/360360 2436559/720720 42142223/12252240 14274301/4084080 275295799/77597520 55835135/15519504 One Hundredth: 14466636279520351160221518043104131447711/2788815009188499086581352357412492142272 (one based) Index of first value: greater than 1: 2 (second term) greater than 2: 4 (fourth term) greater than 3: 11 (eleventh term) greater than 4: 31 (thirty-first term) greater than 5: 83 (eighty-third term) greater than 6: 227 (two hundred and twenty-seventh term) greater than 7: 616 (six hundred and sixteenth term) greater than 8: 1,674 (one thousand, six hundred and seventy-fourth term) greater than 9: 4,550 (four thousand, five hundred and fiftieth term) greater than 10: 12,367 (twelve thousand, three hundred and sixty-seventh term)
(The last two entries are too slow/beyond the reach of the first version of mpfr.js, but it's quick on the desktop)
using standard floats
integer n = 1, gn = 1 atom hn = 1 sequence gt = {} puts(1,"First twenty harmonic numbers as fractions:\n") while gn<=10 do if n<=20 then printf(1,"%18.15f%s",{hn,iff(mod(n,5)?" ","\n")}) end if if n=100 then printf(1,"\nOne Hundredth: %18.15f\n\n",{hn}) end if if hn>gn then gt &= n gn += 1 end if n += 1 hn += 1/n end while printf(1,"(one based) Index of first value:\n") for i=1 to length(gt) do printf(1," greater than %2d: %,6d (%s term)\n",{i,gt[i],ordinal(gt[i])}) end for {} = wait_key()
- Output:
First twenty harmonic numbers as fractions: 1.000000000000000 1.500000000000000 1.833333333333333 2.083333333333333 2.283333333333333 2.450000000000000 2.592857142857143 2.717857142857143 2.828968253968254 2.928968253968254 3.019877344877345 3.103210678210678 3.180133755133755 3.251562326562327 3.318228993228994 3.380728993228994 3.439552522640758 3.495108078196314 3.547739657143682 3.597739657143682 One Hundredth: 5.187377517639621 (one based) Index of first value: greater than 1: 2 (second term) greater than 2: 4 (fourth term) greater than 3: 11 (eleventh term) greater than 4: 31 (thirty-first term) greater than 5: 83 (eighty-third term) greater than 6: 227 (two hundred and twenty-seventh term) greater than 7: 616 (six hundred and sixteenth term) greater than 8: 1,674 (one thousand, six hundred and seventy-fourth term) greater than 9: 4,550 (four thousand, five hundred and fiftieth term) greater than 10: 12,367 (twelve thousand, three hundred and sixty-seventh term)
Prolog
<lang prolog>main:-
print_harmonic_series(20), nl, nth_harmonic_number(100, T), Num is numerator(T), Denom is denominator(T), writef('100th harmonic number: %t/%t\n', [Num, Denom]), nl, print_first_harmonic_greater_than(10).
print_harmonic_series(N):-
writef('First %t harmonic numbers:\n', [N]), harmonic_first(H), print_harmonic_series(N, H).
print_harmonic_series(N, H):-
H = h(I, T), Num is numerator(T), Denom is denominator(T), writef('%3r. %t/%t\n', [I, Num, Denom]), (I == N, ! ; harmonic_next(H, H1), print_harmonic_series(N, H1)).
print_first_harmonic_greater_than(N):-
harmonic_first(H), print_first_harmonic_greater_than(1, N, H).
print_first_harmonic_greater_than(N, L, _):-
N > L, !.
print_first_harmonic_greater_than(N, L, H):-
H = h(P, T), (T > N -> writef('Position of first term >%3r: %t\n', [N, P]), N1 is N + 1 ; N1 = N), harmonic_next(H, H1), print_first_harmonic_greater_than(N1, L, H1).
harmonic_first(h(1, 1)).
harmonic_next(h(N1, T1), h(N2, T2)):-
N2 is N1 + 1, T2 is T1 + 1 rdiv N2.
nth_harmonic_number(N, T):-
harmonic_first(H), nth_harmonic_number(N, T, H).
nth_harmonic_number(N, T, h(N, T)):-!. nth_harmonic_number(N, T, H1):-
harmonic_next(H1, H2), nth_harmonic_number(N, T, H2).</lang>
- Output:
First 20 harmonic numbers: 1. 1/1 2. 3/2 3. 11/6 4. 25/12 5. 137/60 6. 49/20 7. 363/140 8. 761/280 9. 7129/2520 10. 7381/2520 11. 83711/27720 12. 86021/27720 13. 1145993/360360 14. 1171733/360360 15. 1195757/360360 16. 2436559/720720 17. 42142223/12252240 18. 14274301/4084080 19. 275295799/77597520 20. 55835135/15519504 100th harmonic number: 14466636279520351160221518043104131447711/2788815009188499086581352357412492142272 Position of first term > 1: 2 Position of first term > 2: 4 Position of first term > 3: 11 Position of first term > 4: 31 Position of first term > 5: 83 Position of first term > 6: 227 Position of first term > 7: 616 Position of first term > 8: 1674 Position of first term > 9: 4550 Position of first term > 10: 12367
Python
A generator function using fractions: <lang python>from fractions import Fraction
def harmonic_series():
n, h = Fraction(1), Fraction(1) while True: yield h h += 1 / (n + 1) n += 1
if __name__ == '__main__':
from itertools import islice for n, d in (h.as_integer_ratio() for h in islice(harmonic_series(), 20)): print(n, '/', d)</lang>
- Output:
1 / 1 3 / 2 11 / 6 25 / 12 137 / 60 49 / 20 363 / 140 761 / 280 7129 / 2520 7381 / 2520 83711 / 27720 86021 / 27720 1145993 / 360360 1171733 / 360360 1195757 / 360360 2436559 / 720720 42142223 / 12252240 14274301 / 4084080 275295799 / 77597520 55835135 / 15519504
Or alternatively, in terms of itertools.accumulate:
<lang python>Harmonic series
from fractions import Fraction from itertools import accumulate, count, islice from operator import add
- harmonic :: [Fraction]
def harmonic():
Non finite stream of the terms of the Harmonic series. return accumulate( (1 / Fraction(x) for x in count(1)), add )
- ------------------------- TEST -------------------------
- main :: IO ()
def main():
Tests of the harmonic series function
print('First 20 terms of the harmonic series:') print('\n'.join([ showFraction(nd) for nd in islice(harmonic(), 20) ]))
print('\n100th term:') print( showFraction( next(islice(harmonic(), 99, None)) ) )
print() print( 'One-based indices of terms above threshold values:' ) indexedHarmonic = enumerate(harmonic()) print('\n'.join([ next( showFirstLimit(n)(x) for x in indexedHarmonic if n < x[1] ) for n in range(1, 1 + 10) ]))
- ------------------ DISPLAY FORMATTING ------------------
- showFraction :: Fraction -> String
def showFraction(nd):
String representation of the fraction nd. n, d = nd.as_integer_ratio()
return f'{n} / {d}'
- showFirstLimit :: Int -> (Int, Fraction) -> String
def showFirstLimit(n):
Report of 1-based index of first term with a value over n def go(indexedFraction): i = indexedFraction[0]
return f'Term {1 + i} is the first above {n}'
return go
- MAIN ---
if __name__ == '__main__':
main()</lang>
- Output:
First 20 terms of the harmonic series: 1 / 1 3 / 2 11 / 6 25 / 12 137 / 60 49 / 20 363 / 140 761 / 280 7129 / 2520 7381 / 2520 83711 / 27720 86021 / 27720 1145993 / 360360 1171733 / 360360 1195757 / 360360 2436559 / 720720 42142223 / 12252240 14274301 / 4084080 275295799 / 77597520 55835135 / 15519504 100th term: 14466636279520351160221518043104131447711 / 2788815009188499086581352357412492142272 One-based indices of terms above threshold values: Term 2 is the first above 1 Term 4 is the first above 2 Term 11 is the first above 3 Term 31 is the first above 4 Term 83 is the first above 5 Term 227 is the first above 6 Term 616 is the first above 7 Term 1674 is the first above 8 Term 4550 is the first above 9 Term 12367 is the first above 10
Quackery
<lang Quackery> [ $ "bigrat.qky" loadfile ] now!
0 n->v 20 times [ i^ 1+ n->v 1/v v+ 2dup 20 point$ echo$ say " = " 2dup vulgar$ echo$ cr ] 2drop cr 1 temp put 0 n->v 1 [ dup dip [ n->v 1/v v+ temp share n->v 2over v< ] swap if [ temp share echo say " : " dup echo cr 1 temp tally ] temp share 11 < while 1+ again ] temp release drop 2drop</lang>
- Output:
1 = 1/1 1.5 = 3/2 1.83333333333333333333 = 11/6 2.08333333333333333333 = 25/12 2.28333333333333333333 = 137/60 2.45 = 49/20 2.59285714285714285714 = 363/140 2.71785714285714285714 = 761/280 2.82896825396825396825 = 7129/2520 2.92896825396825396825 = 7381/2520 3.01987734487734487734 = 83711/27720 3.10321067821067821068 = 86021/27720 3.18013375513375513376 = 1145993/360360 3.25156232656232656233 = 1171733/360360 3.31822899322899322899 = 1195757/360360 3.38072899322899322899 = 2436559/720720 3.43955252264075793488 = 42142223/12252240 3.49510807819631349043 = 14274301/4084080 3.54773965714368191148 = 275295799/77597520 3.59773965714368191148 = 55835135/15519504 1 : 2 2 : 4 3 : 11 4 : 31 5 : 83 6 : 227 7 : 616 8 : 1674 9 : 4550 10 : 12367
R
The talk page helpfully points out that we can be remarkably lazy here. <lang R>HofN <- function(n) sum(1/seq_len(n)) #Task 1 H <- sapply(1:100000, HofN) print(H[1:20]) #Task 2 print(sapply(1:10, function(x) which.max(H > x))) #Task 3 and stretch</lang>
- Output:
> print(H[1:20]) #Task 2 [1] 1.000000 1.500000 1.833333 2.083333 2.283333 2.450000 2.592857 2.717857 2.828968 2.928968 3.019877 3.103211 3.180134 3.251562 [15] 3.318229 3.380729 3.439553 3.495108 3.547740 3.597740 > print(sapply(1:10, function(x) which.max(H > x))) #Task 3 and stretch [1] 2 4 11 31 83 227 616 1674 4550 12367
Raku
Using Lingua::EN::Numbers from the Raku ecosystem. <lang perl6>use Lingua::EN::Numbers;
my @H = [\+] (1..*).map: { FatRat.new: 1, $_ };
say "First twenty harmonic numbers as rationals:\n",
@H[^20]».&pretty-rat.batch(5)».fmt("%18s").join: "\n";
put "\nOne Hundredth:\n", pretty-rat @H[99];
say "\n(zero based) Index of first value:"; printf " greater than %2d: %6s (%s term)\n",
$_, comma( my $i = @H.first(* > $_, :k) ), ordinal 1 + $i for 1..10;</lang>
- Output:
First twenty harmonic numbers as rationals: 1 3/2 11/6 25/12 137/60 49/20 363/140 761/280 7129/2520 7381/2520 83711/27720 86021/27720 1145993/360360 1171733/360360 1195757/360360 2436559/720720 42142223/12252240 14274301/4084080 275295799/77597520 55835135/15519504 One Hundredth: 14466636279520351160221518043104131447711/2788815009188499086581352357412492142272 (zero based) Index of first value: greater than 1: 1 (second term) greater than 2: 3 (fourth term) greater than 3: 10 (eleventh term) greater than 4: 30 (thirty-first term) greater than 5: 82 (eighty-third term) greater than 6: 226 (two hundred twenty-seventh term) greater than 7: 615 (six hundred sixteenth term) greater than 8: 1,673 (one thousand, six hundred seventy-fourth term) greater than 9: 4,549 (four thousand, five hundred fiftieth term) greater than 10: 12,366 (twelve thousand, three hundred sixty-seventh term)
REXX
The default number of decimal digits (9) could've been used instead of 80 for this task's particular limits. <lang rexx>/*REXX pgm to calculate N numbers (sums) in the harmonic series and also when they > X. */ parse arg digs sums high ints /*obtain optional arguments from the CL*/ if digs= | digs="," then digs= 80 /*Not specified? Then use the default.*/ if sums= | sums="," then sums= 20 /* " " " " " " */ if high= | high="," then high= 10 /* " " " " " " */ if ints= | ints="," then ints= 1 2 3 4 5 6 7 8 9 10 /*Not specified? " " " */ w= length(sums) + 2 /*width of Nth harmonic index + suffix.*/ numeric digits digs /*have REXX use more numeric dec. digs.*/
s= 0 /*initialize harmonic series sum to 0. */ do j=1 for sums; s= s + 1/j /*calc "sums" of harmonic series nums.*/ @iter= right((j)th(j), w) /*obtain a nicely formatted sum index. */ say right(@iter, w) 'harmonic sum ──►' s /*indent the output to the terminal. */ end /*j*/
say /*have a blank line between output sets*/ many= words(ints) /*obtain number of limits to be used. */ z= word(ints, 1) /* " the first " " " " */ lastInt= word(ints, many) /* " " last " " " " */ w= length(lastInt) /*W: is the maximum width of any limit*/
- = 1 /*a pointer to a list of integer limits*/
s= 0 /*initialize harmonic series sum to 0. */ do j=1; s= s + 1/j /*calculate sums of harmonic sum index.*/ if s<=z then iterate /*Is sum <= a limit? Then keep going. */ iter= commas(j)th(j) /*obtain a nicely formatted sum index. */ L= length(iter) /*obtain length so as to align output. */ @iter= right(iter, max(L, 25) ) /*indent the output to the terminal. */ say @iter " iteration of the harmonic series, the sum is greater than " right(z, w) #= # + 1 /*bump the pointer to the next limit. */ if #>many then leave /*Are at the end of the limits? Done. */ z= word(ints, #) /*point to the next limit to be used. */ end /*j*/ /* [↑] above indices are unity─based. */
exit 0 /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ commas: parse arg ?; do jc=length(?)-3 to 1 by -3; ?=insert(',', ?, jc); end; return ? th: parse arg x; return word('th st nd rd', 1 + (x//10) *(x//100%10\==1) *(x//10<4))</lang>
- output when using the default inputs:
1st harmonic sum ──► 1 2nd harmonic sum ──► 1.5 3rd harmonic sum ──► 1.8333333333333333333333333333333333333333333333333333333333333333333333333333333 4th harmonic sum ──► 2.0833333333333333333333333333333333333333333333333333333333333333333333333333333 5th harmonic sum ──► 2.2833333333333333333333333333333333333333333333333333333333333333333333333333333 6th harmonic sum ──► 2.4500000000000000000000000000000000000000000000000000000000000000000000000000000 7th harmonic sum ──► 2.5928571428571428571428571428571428571428571428571428571428571428571428571428571 8th harmonic sum ──► 2.7178571428571428571428571428571428571428571428571428571428571428571428571428571 9th harmonic sum ──► 2.8289682539682539682539682539682539682539682539682539682539682539682539682539682 10th harmonic sum ──► 2.9289682539682539682539682539682539682539682539682539682539682539682539682539682 11th harmonic sum ──► 3.0198773448773448773448773448773448773448773448773448773448773448773448773448773 12th harmonic sum ──► 3.1032106782106782106782106782106782106782106782106782106782106782106782106782106 13th harmonic sum ──► 3.1801337551337551337551337551337551337551337551337551337551337551337551337551337 14th harmonic sum ──► 3.2515623265623265623265623265623265623265623265623265623265623265623265623265623 15th harmonic sum ──► 3.3182289932289932289932289932289932289932289932289932289932289932289932289932290 16th harmonic sum ──► 3.3807289932289932289932289932289932289932289932289932289932289932289932289932290 17th harmonic sum ──► 3.4395525226407579348755819344054638172285231108760520525226407579348755819344055 18th harmonic sum ──► 3.4951080781963134904311374899610193727840786664316076080781963134904311374899611 19th harmonic sum ──► 3.5477396571436819114837690689083877938367102453789760291308278924377995585425927 20th harmonic sum ──► 3.5977396571436819114837690689083877938367102453789760291308278924377995585425927 2nd iteration of the harmonic series, the sum is greater than 1 4th iteration of the harmonic series, the sum is greater than 2 11th iteration of the harmonic series, the sum is greater than 3 31st iteration of the harmonic series, the sum is greater than 4 83rd iteration of the harmonic series, the sum is greater than 5 227th iteration of the harmonic series, the sum is greater than 6 616th iteration of the harmonic series, the sum is greater than 7 1,674th iteration of the harmonic series, the sum is greater than 8 4,550th iteration of the harmonic series, the sum is greater than 9 12,367th iteration of the harmonic series, the sum is greater than 10
Wren
<lang ecmascript>import "/big" for BigRat import "/fmt" for Fmt
var harmonic = Fn.new { |n| (1..n).reduce(BigRat.zero) { |sum, i| sum + BigRat.one/i } }
BigRat.showAsInt = true System.print("The first 20 harmonic numbers and the 100th, expressed in rational form, are:") var numbers = (1..20).toList numbers.add(100) for (i in numbers) Fmt.print("$3d : $s", i, harmonic.call(i))
System.print("\nThe first harmonic number to exceed the following integers is:") var i = 1 var limit = 10 var n = 1 var h = 0 while (true) {
h = h + 1/n if (h > i) { Fmt.print("integer = $2d -> n = $,6d -> harmonic number = $9.6f (to 6dp)", i, n, h) i = i + 1 if (i > limit) return } n = n + 1
}</lang>
- Output:
The first 20 harmonic numbers and the 100th, expressed in rational form, are: 1 : 1 2 : 3/2 3 : 11/6 4 : 25/12 5 : 137/60 6 : 49/20 7 : 363/140 8 : 761/280 9 : 7129/2520 10 : 7381/2520 11 : 83711/27720 12 : 86021/27720 13 : 1145993/360360 14 : 1171733/360360 15 : 1195757/360360 16 : 2436559/720720 17 : 42142223/12252240 18 : 14274301/4084080 19 : 275295799/77597520 20 : 55835135/15519504 100 : 14466636279520351160221518043104131447711/2788815009188499086581352357412492142272 The first harmonic number to exceed the following integers is: integer = 1 -> n = 2 -> harmonic number = 1.500000 (to 6dp) integer = 2 -> n = 4 -> harmonic number = 2.083333 (to 6dp) integer = 3 -> n = 11 -> harmonic number = 3.019877 (to 6dp) integer = 4 -> n = 31 -> harmonic number = 4.027245 (to 6dp) integer = 5 -> n = 83 -> harmonic number = 5.002068 (to 6dp) integer = 6 -> n = 227 -> harmonic number = 6.004367 (to 6dp) integer = 7 -> n = 616 -> harmonic number = 7.001274 (to 6dp) integer = 8 -> n = 1,674 -> harmonic number = 8.000486 (to 6dp) integer = 9 -> n = 4,550 -> harmonic number = 9.000208 (to 6dp) integer = 10 -> n = 12,367 -> harmonic number = 10.000043 (to 6dp)