A004080 -id:A004080 - cp's OEIS frontend

A242654 1 followed by the union of the terms > 2 in A002387 (or A004080) and A115515.

1, 3, 4, 10, 11, 30, 31, 82, 83, 226, 227, 615, 616, 1673, 1674, 4549, 4550, 12366, 12367, 33616, 33617, 91379, 91380, 248396, 248397, 675213, 675214, 1835420, 1835421, 4989190, 4989191, 13562026, 13562027, 36865411, 36865412, 100210580, 100210581, 272400599, 272400600, 740461600, 740461601, 2012783314

Offset: 1

N. J. A. Sloane, May 29 2014

nonn

Ray Chandler, May 29 2014, proposes this as the most likely continuation of A079353.

Cf. A002387, A004080, A115515, A079353.

b[n_] := Ceiling[k /. FindRoot[HarmonicNumber[k] == n, {k, Exp[n]}, WorkingPrecision -> 100]] - 1;
bb = Array[b, 22];
A242654 = Union[bb, bb + 1] // Rest (* Jean-François Alcover, Apr 10 2019 *)

A194541 Partial sums of A004080.

Original entry on oeis.org

1, 5, 16, 47, 130, 357, 973, 2647, 7197, 19564, 53181, 144561, 392958, 1068172, 2903593, 7892784, 21454811, 58320223, 158530804, 430931404, 1171393005, 3184176320, 8655488630, 23528057461, 63955891057, 173850136486, 472573666887, 1284588411309

Offset: 1

Joseph Foley, Aug 28 2011

nonn

The ratio of a(n) to A004080(n+1) converges to e/(e-1), which is approximately equal to 1.581976706. For example, a(21)/A004080(22) = 1171393005/740461601 = 1.581976706716...

Ronald E. Bruck, Maths_Exam_Solns (see page 6).

Cf. A004080.

A002387 Least k such that H(k) > n, where H(k) is the harmonic number Sum_{i=1..k} 1/i.

Original entry on oeis.org

1, 2, 4, 11, 31, 83, 227, 616, 1674, 4550, 12367, 33617, 91380, 248397, 675214, 1835421, 4989191, 13562027, 36865412, 100210581, 272400600, 740461601, 2012783315, 5471312310, 14872568831, 40427833596, 109894245429, 298723530401, 812014744422

Offset: 0

N. J. A. Sloane

From Dean Hickerson, Apr 19 2003: (Start)
For k >= 1, log(k + 1/2) + gamma < H(k) < log(k + 1/2) + gamma + 1/(24k^2), where gamma is Euler's constant (A001620). It is likely that the upper and lower bounds have the same floor for all k >= 2, in which case a(n) = floor(exp(n-gamma) + 1/2) for all n >= 0.
This remark is based on a simple heuristic argument. The lower and upper bounds differ by 1/(24k^2), so the probability that there's an integer between the two bounds is 1/(24k^2). Summing that over all k >= 2 gives the expected number of values of k for which there's an integer between the bounds. That sum equals Pi^2/144 - 1/24 ~ 0.02687. That's much less than 1, so it is unlikely that there are any such values of k.
(End)
Referring to A118050 and A118051, using a few terms of the asymptotic series for the inverse of H(x), we can get an expression which, with greater likelihood than mentioned above, should give a(n) for all n >= 0. For example, using the same type of heuristic argument given by Dean Hickerson, it can be shown that, with probability > 99.995%, we should have, for all n >= 0, a(n) = floor(u + 1/2 - 1/(24u) + 3/(640u^3)) where u = e^(n - gamma). - David W. Cantrell (DWCantrell(AT)sigmaxi.net)
For k > 1, H(k) is never an integer. Hence apart from the first two terms this sequence coincides with A004080. - Nick Hobson, Nov 25 2006

John H. Conway and R. K. Guy, "The Book of Numbers," Copernicus, an imprint of Springer-Verlag, NY, 1996, pages 258-259.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 83, p. 28, Ellipses, Paris 2008.
Ronald Lewis Graham, Donald Ervin Knuth and Oren Patashnik, "Concrete Mathematics, a Foundation for Computer Science," Addison-Wesley Publishing Co., Reading, MA, 1989, Page 258-264, 438.
H. P. Robinson, Letter to N. J. A. Sloane, Oct 23 1973.
W. Sierpiński, Sur les decompositions de nombres rationnels, Oeuvres Choisies, Académie Polonaise des Sciences, Warsaw, Poland, 1974, p. 181.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane, Illustration for sequence M4299 (=A007340) in The Encyclopedia of Integer Sequences (with Simon Plouffe), Academic Press, 1995.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
I. Stewart, L'univers des nombres, pp. 54, Belin-Pour La Science, Paris 2000.

Robert G. Wilson v, Table of n, a(n) for n = 0..2303 (first 101 terms from T. D. Noe)
John V. Baxley, Euler's constant, Taylor's formula, and slowly converging series, Math. Mag. 65 (1992), 302-313. (Gives terms up to n = 24.)
R. P. Boas, Partial sums of the harmonic series, II, Mimeographed manuscript, no date.
R. P. Boas, Jr. and J. W. Wrench, Jr., Partial sums of the harmonic series, Amer. Math. Monthly, 78 (1971), 864-870. (Gives terms up to n = 20.)
Nick Hobson, Solution to puzzle 34: Harmonic sum 2.
H. P. Robinson, Letter to N. J. A. Sloane, Oct 28 1973.
H. P. Robinson, Letter to N. J. A. Sloane, Oct 1981
H. P. Robinson, Letter to N. J. A. Sloane, Dec 20 1983
John A. Rochowicz, Jr., Harmonic Numbers: Insights, Approximations and Applications, Spreadsheets in Education (eJSiE) (2015), Vol. 8: Iss. 2, Article 4.
R. G. Wilson, Letter to N. J. A. Sloane with attachment, Jan 1989 (A006509 is mentioned in the attachment)
R. G. Wilson v, Letter to N. J. A. Sloane, Oct 12 1993
J. W. Wrench, Jr., Selected Partial Sums of the Harmonic Series, Manuscript, no date [Annotated scanned copy]

Apart from initial terms, same as A004080.

Cf. A006509, A055980, A115515, A242654.

a002387 n = a002387_list !! n
a002387_list = f 0 1 where
   f x k = if hs !! k > fromIntegral x
           then k : f (x + 1) (k + 1) else f x (k + 1)
           where hs = scanl (+) 0 $ map recip [1..]
-- Reinhard Zumkeller, Aug 04 2014

fh[0]=0; fh[1]=1; fh[k_] := Module[{tmp}, If[Floor[tmp=Log[k+1/2]+EulerGamma]==Floor[tmp+1/(24k^2)], Floor[tmp], UNKNOWN]]; a[0]=1; a[1]=2; a[n_] := Module[{val}, val=Round[Exp[n-EulerGamma]]; If[fh[val]==n&&fh[val-1]==n-1, val, UNKNOWN]]; (* fh[k] is either floor(H(k)) or UNKNOWN *)
f[n_] := k /. FindRoot[HarmonicNumber[k] == n, {k, Exp[n]}, WorkingPrecision -> 100] // Ceiling; f[0] = 1; Array[f, 28, 0] (* Robert G. Wilson v, Jan 24 2017 after Jean-François Alcover in A014537 *)

a(n)=if(n,my(k=exp(n-Euler));ceil(solve(x=k-1.5,k+.5,intnum(y=0,1,(1-y^x)/(1-y))-n)),1) \\ Charles R Greathouse IV, Jun 13 2012

Note that the conditionally convergent series Sum_{k>=1} (-1)^(k+1)/k = log 2 (A002162).
Limit_{n->oo} a(n+1)/a(n) = e. - Robert G. Wilson v, Dec 07 2001
It is conjectured that, for n > 1, a(n) = floor(exp(n-gamma) + 1/2). - Benoit Cloitre, Oct 23 2002

Terms for n >= 13 computed by Eric W. Weisstein; corrected by James R. Buddenhagen and Eric W. Weisstein, Feb 18 2001

Edited by Dean Hickerson, Apr 19 2003

A046024 a(n) = smallest k such that Sum_{ i = 1..k } 1/prime(i) exceeds n.

Original entry on oeis.org

1, 3, 59, 361139, 43922730588128390

Offset: 0

Eric W. Weisstein

The corresponding primes prime(a(n)) are in A016088.

Index m for which the prime harmonic number p[ m ] := Sum[ 1/Prime[ k ],{k,1,m} ] >= n.

E. Bach, D. Klyve, and J. P. Sorenson, Computing prime harmonic sums, Math. Comp. 78 (2009) 2283-2305.
Eric Weisstein's World of Mathematics, Prime Number.
Eric Weisstein's World of Mathematics, Harmonic Series of Primes.

Cf. A004080, A016088, A096232, A223037.

Cf. A024451/A002110 for Sum_{i = 1..n} 1/prime(i).

Table[m = 1; s = 0; While[(s = s + 1/Prime[m]) <= n, m++];
m, {n, 0, 4}] (* Robert Price, Mar 27 2019 *)

a(n)=my(t); forprime(p=2,, t+=1./p; if(t>n, return(primepi(p)))) \\ Charles R Greathouse IV, Apr 29 2015

From Jonathan Sondow, Apr 17 2013: (Start)
a(n) = A000720(A016088(n)) = A000720(A096232(n))+1.
a(n) = e^(e^(n + O(1))), see comment in A223037. [corrected by Charles R Greathouse IV, Aug 22 2013] (End)
a(n) = A103591(2*n). - Michel Marcus, Aug 22 2013

a(4) found by Tomás Oliveira e Silva (tos(AT)det.ua.pt), using the fourth term of A016088. - Dec 14 2005

a(0) from Jonathan Sondow, Apr 16 2013

A019529 Sum of a(n) terms of 1/sqrt(k) first strictly exceeds n.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 14, 18, 22, 27, 33, 39, 45, 52, 60, 68, 76, 85, 95, 105, 115, 126, 138, 150, 162, 175, 189, 202, 217, 232, 247, 263, 280, 297, 314, 332, 351, 370, 389, 409, 430, 451, 472, 494, 517, 540, 563, 587, 612, 637, 662, 688, 715, 741, 769, 797, 825

Offset: 0

Robert G. Wilson v

nonn

			Let b(k) = 1 + 1/sqrt(2) + 1/sqrt(3) + ... + 1/sqrt(k):
.k.......1....2.....3.....4.....5.....6.....7
-------------------------------------------------
b(k)...1.00..1.71..2.28..2.78..3.23..3.64..4.01
For A019529 we have:
  n=0: smallest k is a(0) = 1 since 1.00 > 0
  n=1: smallest k is a(1) = 2 since 1.71 > 1
  n=2: smallest k is a(2) = 3 since 2.28 > 2
  n=3: smallest k is a(3) = 5 since 3.23 > 3
  n=4: smallest k is a(4) = 7 since 4.01 > 4
For A054040 we have:
  n=1: smallest k is a(1) = 1 since 1.00 >= 1
  n=2: smallest k is a(2) = 3 since 2.28 >= 2
  n=3: smallest k is a(3) = 5 since 3.23 >= 3
  n=4: smallest k is a(4) = 7 since 4.01 >= 4

A054040 is another version. See also A002387, A004080.

s = 0; k = 1; Do[ While[ s <= n, s = s + N[ 1/Sqrt[ k ], 24 ]; k++ ]; Print[ k - 1 ], {n, 1, 75} ]
With[{c=N[Accumulate[Table[1/Sqrt[x],{x,1000}]]]},Table[Position[c,?(#>n&),1,1],{n,0,1000}]]//Flatten (* _Harvey P. Dale, Mar 19 2025 *)

Edited by N. J. A. Sloane, Sep 01 2009

A056903 Numbers n such that the numerator of the rational number 1 + 1/2 + 1/3 + ... + 1/n is a prime number.

Original entry on oeis.org

2, 3, 5, 8, 9, 21, 26, 41, 56, 62, 69, 79, 89, 91, 122, 127, 143, 167, 201, 230, 247, 252, 290, 349, 376, 459, 489, 492, 516, 662, 687, 714, 771, 932, 944, 1061, 1281, 1352, 1489, 1730, 1969, 2012, 2116, 2457, 2663, 2955, 3083, 3130, 3204, 3359, 3494, 3572

Offset: 1

James R. Buddenhagen, Feb 23 2001

nonn

Related to partial sums of the harmonic series and to Wolstenholme's Theorem.

Some of the larger entries may only correspond to probable primes.

			5 is in this sequence because 1+1/2+1/3+1/4+1/5 = 137/60 and 137 is prime.

Eric Weisstein, Table of n, a(n) for n = 1..97
J. Sondow and E. W. Weisstein, MathWorld: Harmonic Number
Eric Weisstein's World of Mathematics, Integer Sequence Primes

Cf. A002387, A004080.

Cf. A001008 (numerator of the harmonic number H(n)), A067657 (primes that are the numerator of a harmonic number).

Select[Range[1000], PrimeQ[Numerator[HarmonicNumber[ # ]]] &]

isok(n) = isprime(numerator(sum(k=1, n, 1/k))); \\ Michel Marcus, Feb 05 2016

use ntheory ":all"; for (1..1000) { say if is_prime((harmfrac($))[0]); } # _Dana Jacobsen, Feb 05 2016

Terms from 201 to 492 computed by Jud McCranie.
More terms from Kamil Duszenko (kdusz(AT)wp.pl), Jun 22 2003
29 more terms from T. D. Noe, Sep 15 2004
Further terms found by Eric W. Weisstein, Mar 07 2005, Mar 29 2005, Nov 28 2005, Sep 23 2006

A055980 a(n) = floor(Sum_{i=1..n} 1/i).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Henry Bottomley, Jul 20 2000

nonn

If we choose at random (uniformly) a permutation in the symmetric group S_n then a(n) is the expected number of cycles (rounded down) in the cycle decomposition of the permutation. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Oct 17 2001

a(n) = A214075(n,n-1) for n > 0. - Reinhard Zumkeller, Jul 03 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
L. D. Kudryavtsev, Harmonic series, The Encyclopedia of Mathematics.
Eric Weisstein's World of Mathematics, High-Water Mark

Cf. A002387, A004080 (indices of records).

import Data.Ratio ((%), denominator)
a055980 = floor . sum . map (1 %) . enumFromTo 1
a055980_list = map floor $ scanl1 (+) $ map (1 %) [1..]
-- Reinhard Zumkeller, Jul 03 2012

Floor[HarmonicNumber[Range[110]]] (* Harvey P. Dale, May 22 2021 *)

a(n) ~ log(n) - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 25 2001

a(n) = floor[A001008(n)/A002805(n)]. - Lekraj Beedassy, Sep 17 2006

A074631 a(n) is the smallest k such that the sum of the first k terms of the composite-harmonic series, Sum_{j=1..k} 1/(j-th composite), is > n.

Original entry on oeis.org

9, 44, 168, 587, 1940, 6192, 19285, 59010, 178122, 531923, 1574706, 4628338, 13521477, 39299115, 113712434, 327752962, 941457955, 2696114317, 7700146599, 21938239766

Offset: 1

Labos Elemer, Aug 27 2002

			1/4 + 1/6 + 1/8 + 1/9 + 1/10 + 1/12 + 1/14 + 1/15 + 1/16 = 1045/1008, but if 1/16 is not present, the sum is less than 1; 16 is the ninth composite number, so a(1) = 9.

Cf. A002387, A002808, A004080, A016088, A046024, A065855, A076751.

NextComposite[n_] := Block[{k = n + 1}, While[PrimeQ[k], k++ ]; k]; s=0; k = 4; Do[While[s = s + 1/k; s < n, k = NextComposite[k]]; Print[k - PrimePi[k] - 1]; k = NextComposite[k], {n, 1, 20}]
Table[Position[Accumulate[1/Select[Range[5*10^6],CompositeQ]],?(#>n&),1,1],{n,12}]//Flatten (* The program generates the first 12 terms of the sequence. *) (* _Harvey P. Dale, Jan 22 2023 *)

lista(cmax) = {my(n = 1, s = 0, k = 0); forcomposite(c = 1, cmax, k++; s += 1/c; if(s > n, print1(k, ", "); n++));} \\ Amiram Eldar, Jul 17 2024

a(n) = Min { k : Sum_{j=1..k} 1/A002808(j) > n }.
Limit_{n->oo} a(n+1)/a(n) = e. - Robert G. Wilson v, Aug 28 2002
a(n) = A065855(A076751(n)). - Amiram Eldar, Jul 17 2024

Edited by Robert G. Wilson v, Aug 28 2002
More terms from Robert Gerbicz, Aug 30 2002
2 more terms from Robert G. Wilson v, Sep 03 2002
Edited by Jon E. Schoenfield, Sep 13 2023
a(18)-a(20) from Amiram Eldar, Jul 17 2024

A115515 a(n) = largest m such that the harmonic number H(m)= Sum_{i=1..m} 1/i is < n.

Original entry on oeis.org

0, 3, 10, 30, 82, 226, 615, 1673, 4549, 12366, 33616, 91379, 248396, 675213, 1835420, 4989190, 13562026, 36865411, 100210580, 272400599, 740461600, 2012783314, 5471312309, 14872568830, 40427833595, 109894245428

Offset: 1

Artur Jasinski, Jan 23 2006

nonn

H. P. Robinson, Letter to N. J. A. Sloane, Oct 1981

Apart from the initial values, this is simply A002387(n)-1. Cf. A004080.

c:=0: H[0]:=0: for n from 1 to 10^4 do H[n]:=1/n+H[n-1]: if floor(H[n])-floor(H[n-1])=1 then c:=1+c: b[c]:=n-1: else c:=c: fi: od: seq(b[j],j=1..c); # Emeric Deutsch

a[n_] := Ceiling[k /. FindRoot[HarmonicNumber[k] == n, {k, Exp[n]}, WorkingPrecision -> 100]] - 1;
Array[a, 26] (* Jean-François Alcover, Apr 10 2019 *)

A136616 a(n) = largest m with H(m) - H(n) <= 1, where H(i) = Sum_{j=1 to i} 1/j, the i-th harmonic number, H(0) = 0.

Original entry on oeis.org

1, 3, 6, 9, 11, 14, 17, 19, 22, 25, 28, 30, 33, 36, 38, 41, 44, 47, 49, 52, 55, 57, 60, 63, 66, 68, 71, 74, 76, 79, 82, 85, 87, 90, 93, 96, 98, 101, 104, 106, 109, 112, 115, 117, 120, 123, 125, 128, 131, 134, 136, 139, 142, 144, 147, 150, 153, 155, 158, 161, 163, 166

Offset: 0

Rainer Rosenthal, Jan 13 2008

			a(3) = 9 because H(9) - H(3) = 1/4 + ... + 1/9 < 1 < 1/4 + ... + 1/10 = H(10) - H(3).

E. R. Bobo, A sequence related to the harmonic series, College Math. J. 26 (1995), 308-310.

Cf. A001008, A002805, A002387, A004080, A079353, A096618, A115515, A014537, A055980, A103762, A136617.

e:= exp(1):
A136616 := n -> floor( e*n + (e-1)/2 + (e - 1/e)/(24*(n + 1/2))):
seq(A136616(n), n=0..50);

default(realprecision, 10^5); e=exp(1);
a(n) = floor(e*n+(e-1)/2+(e-1/e)/(24*n+12)); \\ Jinyuan Wang, Mar 06 2020

a(n) = floor(e*n + (e-1)/2 + (e - 1/e)/(24*(n + 1/2))), after a suggestion by David Cantrell.

a(n) = A103762(n+1) - 1 = A136617(n+1) + n for n > 0. - Jinyuan Wang, Mar 06 2020

Definition corrected by David W. Cantrell, Apr 14 2008

cp's OEIS Frontend

A242654 1 followed by the union of the terms > 2 in A002387 (or A004080) and A115515.

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A194541 Partial sums of A004080.

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A002387 Least k such that H(k) > n, where H(k) is the harmonic number Sum_{i=1..k} 1/i.

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A046024 a(n) = smallest k such that Sum_{ i = 1..k } 1/prime(i) exceeds n.

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A019529 Sum of a(n) terms of 1/sqrt(k) first strictly exceeds n.

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A056903 Numbers n such that the numerator of the rational number 1 + 1/2 + 1/3 + ... + 1/n is a prime number.

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A055980 a(n) = floor(Sum_{i=1..n} 1/i).

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A074631 a(n) is the smallest k such that the sum of the first k terms of the composite-harmonic series, Sum_{j=1..k} 1/(j-th composite), is > n.

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