A056053 -id:A056053 - cp's OEIS frontend

A092315 a(n) is the smallest m such that the partial sum of the odd harmonic series Sum_{j=0..m} 1/(2j+1) is > n.

1, 7, 56, 418, 3091, 22845, 168803, 1247297, 9216353, 68100150, 503195828, 3718142207, 27473561357, 203003686105, 1500005624923, 11083625711270, 81897532160124, 605145459495140, 4471453748222756, 33039822589391675, 244133102611731230, 1803913190804074903

Offset: 1

N. J. A. Sloane, Feb 16 2004

nonn

From Gerhard Kirchner, May 21 2020: (Start)
The terms a(n), evaluated by the formula, should pass the test OH(a(n))=n and OH(a(n)-1)=n-1, where OH(m) is the odd harmonic series, see above.
Another formula, see link Asymptotic formulas, formula 1, is OH(m) = (log(4*m)+gamma)/2+1/(2*m)-11/(48*m^2)+1/(8*m^3)-127*t/(1920*m^4), 0The Maxima code includes both tests and creates a b-file in the current directory. For n<=1000, the case "Precision too low" does not occur. (End)
a(2) = 7 and a(3) = 56 are related to the Borwein integrals. Concretely, a(2) = 7 is the smallest m such that the integral Integral_{x=-oo..oo} Product_{k=0..m} (sin((2*k+1)*x)/((2*k+1)*x)) dx is slightly less than Pi, and a(3) = 56 is the smallest m such that the integral Integral_{x=-oo..oo} cos(x) * Product_{k=0..m} (sin((2*k+1)*x)/((2*k+1)*x)) dx is slightly less than Pi/2. See the Wikipedia link and the 3Blue1Brown video link below. - Jianing Song, Dec 10 2022

Gerhard Kirchner, Table of n, a(n) for n = 1..1000
Gerhard Kirchner, Asymptotic formulas
Grant Sanderson, Researchers thought this was a bug (Borwein integrals), 3Blue1Brown video (2022).
Wikipedia, Borwein integral

Except for first term, same as A092318. Equals (A056053-1)/2.

Cf. A074599, A025547, A281355.

A092315[n_] := Floor[Exp[2*n - EulerGamma]/4]; Table[A092315[n], {n, 1, 22}] (* Robert P. P. McKone, Jul 13 2021 *)

block(
fpprec:1000, gam: %gamma, nmax:1000,
fl: openw("bfile1000.txt"),
OH(k,t):=(log(4*k)+gam)/2+1/(2*k)-11/(48*k^2)+1/(8*k^3)-127*t/(1920*k^4),
printf(fl, "1 1"),   newline(fl),
for n from 2 thru nmax do
(u: bfloat(exp(2*n-gam)/4), k: floor(u),
x0: bfloat(OH(k,0)), x01: bfloat(OH(k,1)), x1: bfloat(OH(k-1,0)),
n0: floor(x0), n01: floor(x01), n1: floor(x1),  m: n,
if n0=n and n01=n and n1=n-1 then
         (h: concat(n, " ", k), printf(fl, h),  newline(fl)) else n: nmax),
if mGerhard Kirchner, Jul 23 2020 */
/* The first nmax terms are saved as a b-file */

a(n) = floor(exp(2*n-gamma)/4+1/8) for all n >= 1 (conjectured; see also comments in A002387). - M. F. Hasler, Jan 22 2017

a(n) = floor(exp(2*n-gamma)/4). - Gerhard Kirchner, Jul 23 2020

More terms from M. F. Hasler, Jan 24 2017

a(17) in the data section and 127 terms in the b-file corrected by Gerhard Kirchner, Jul 23 2020

A056054 a(n) = smallest even number 2m such that value of odd harmonic series Sum_{j=0..m} 1/(2j) is > n.

Original entry on oeis.org

8, 62, 454, 3348, 24734, 182760, 1350428, 9978382, 73730824, 544801200, 4025566630, 29745137662, 219788490858, 1624029488844, 12000044999386, 88669005690160, 655180257281000, 4841163675961122, 35771629985782052

Offset: 1

Robert G. Wilson v, Jul 25 2000 and Jan 11 2004

nonn

Numbers 2*m such that floor(f(m)) = floor(f(m-1)) where f(m) = Sum_{j=1..m} ((2*j-1)/(2*j)). Examples:
  floor(f(1))=floor(1/2)=0;
  floor(f(2))=floor(1/2+3/4)=floor(1.25)=1, then 2*2=4 is not in the sequence;
  floor(f(3))=floor(1/2+3/4+5/6)=floor(2.083..)=2, then 2*3=6 is not in the sequence;
  floor(f(4))=floor(1/2+3/4+5/6+7/8)=floor(2.958..)=2, then 2*4=8 is the first term of the sequence. - Philippe Lallouet (philip.lallouet(AT)wanadoo.fr), Aug 15 2007

Calvin C. Clawson, Mathematical Mysteries, The Beauty and Magic of Numbers, Plenum Press, NY and London, 1996, page 64.

Cf. A002387, A056053, A091463, A091464, A091465.

Cf. A056054.

s = 0; k = 2; Do[ While[s = N[s + 1/k, 24]; s <= n, k += 2]; Print[k]; k += 2, {n, 1, 12}]
(* or assuming that the Mathematica coding in A002387 is correct then *)
b[n_] := Module[{k = Floor[2a[2n]]}, If[ EvenQ[k], k, k + 1]]; Table[ b[n], {n, 19}] (* Robert G. Wilson v, Apr 17 2004 *)

a(n) = 2*A002387(2n).

The next term is approximately the previous term * e^2.

A281355 a(n) = A092318(n) + 1: Number of terms of the odd harmonic series 1 + 1/3 + 1/5 + 1/7 + ... required to reach a sum >= n.

Original entry on oeis.org

1, 8, 57, 419, 3092, 22846, 168804, 1247298, 9216354, 68100151, 503195829, 3718142208, 27473561358, 203003686106, 1500005624924, 11083625711271, 81897532160125, 605145459495141, 4471453748222757, 33039822589391676, 244133102611731231, 1803913190804074904, 13329215764452299411

Offset: 1

N. J. A. Sloane, Jan 22 2017, following a suggestion from Jerry Polfer

nonn

A different way of listing the sums mentioned in A092318.

A092318 is the main entry for this problem.

Gerhard Kirchner, Table of n, a(n) for n = 1..1000 [replacing an older version by Vincenzo Librandi]
Jerry V Polfer, found a new sequence, not sure if interesting, and how to properly submit, Seqfan list, Jan. 22, 2017 (and follow-up messages).
Albert Säfström, n describes the expected number of loops created by tieing together two random loose ends of a(n) ropes until there is none left.

Cf. A002387, A092318.

a[n_]:=Floor[Exp[2*n-EulerGamma]/4+1]-Boole[n==1]; Array[a,23] (* Stefano Spezia, Jun 25 2024 *)

Lim_{n -> oo} a(n)/exp(2*n) = 1/4e^gamma ~ 0.140364870891721292456...;

a(n) = floor(exp(2*n-gamma)/4+1), for all given values a(n) > 1. - M. F. Hasler and Robert G. Wilson v, Jan 23 2017 [corrected by Gerhard Kirchner, Jul 25 2020]

More explicit, self-contained definition by M. F. Hasler, Jan 22 2017
More terms (computed using A056053) from M. F. Hasler, Jan 23 2017
a(17) corrected in data and 127 terms in the b-file, according to the corrections in A092315, Gerhard Kirchner, Jul 27 2020

A091463 a(n) is the smallest j such that 1/1 + 1/4 + 1/7 + ... + 1/j exceeds n.

Original entry on oeis.org

4, 52, 1060, 21301, 427873, 8594032, 172615738, 3467079760, 69638158519

Offset: 1

Robert G. Wilson v, Jan 12 2004

Cf. A002387, A056053, A056054, A091462, A091464.

s = 0; k = 1; Do[ While[s = N[s + 1/k, 24]; s <= n, k += 3]; Print[k]; k += 3, {n, 1, 7}]

The next term is approximately the previous term * e^3.

Name edited by Jon E. Schoenfield, Dec 20 2019

a(8)-a(9) from Hugo Pfoertner, Dec 27 2019

A091464 a(n) is the smallest j such that 1/2 + 1/5 + 1/8 + ... + 1/j exceeds n.

Original entry on oeis.org

17, 323, 6506, 130664, 2624438, 52713275, 1058774426, 21266052797, 427140088670

Offset: 1

Robert G. Wilson v, Jan 12 2004

Cf. A002387, A056053, A056054, A091462, A091463.

s = 0; k = 2; Do[ While[s = N[s + 1/k, 24]; s <= n, k += 3]; Print[k]; k += 3, {n, 1, 7}]

The next term is approximately the previous term * e^3.

a(8)-a(9) from Hugo Pfoertner, Dec 26 2019

A092317 a(n) = smallest odd number 2m+1 such that the partial sum Sum_{j=0..m} 1/(2j+1) of the odd harmonic series is >= n.

Original entry on oeis.org

1, 15, 113, 837, 6183, 45691, 337607, 2494595, 18432707, 136200301, 1006391657, 7436284415, 54947122715, 406007372211, 3000011249847, 22167251422541, 163795064320249, 1210290918990281, 8942907496445513, 66079645178783351

Offset: 1

N. J. A. Sloane, Feb 16 2004

nonn

Except for first term, same as A056053. Equals 2*A092318 + 1. Cf. A074599, A025547

a(n) ~ C*exp(2n) with C = 0.2807297417834425... - M. F. Hasler, Jan 22 2017

More terms (via A056053) from M. F. Hasler, Jan 22 2017

a(17) corrected - see correction in A092315. Gerhard Kirchner, Jul 25 2020

A091462 a(n) is the smallest j such that 1/3 + 1/6 + 1/9 + ... + 1/j exceeds n.

Original entry on oeis.org

3, 33, 681, 13650, 274140, 5506263, 110596236, 2221384803, 44617706493, 896170591203, 18000067499079, 361541020372644, 7261745513941683, 145856057647068072, 2929597231340774769, 58842533360163495285, 1181883876459465987195, 23738772239546776075803, 476805986328559173414774

Offset: 0

Robert G. Wilson v, Jan 12 2004

nonn

Cf. A002387, A056053, A056054, A091463, A091464.

s = 0; k = 3; Do[ While[s = N[s + 1/k, 24]; s <= n, k += 3]; Print[k]; k += 3, {n, 1, 12}]

a(n) = 3*A002387(3n).

The next term is approximately the previous term * e^3.

a(0) prepended and more terms added by Max Alekseyev, Sep 01 2023

A092267 Values 2m_0+1 = 1, 2m_1, 2m_2+1, ... associated with divergent series T shown below.

Original entry on oeis.org

1, 454, 45891, 547208496, 3013267310449, 1961694770407970734, 589785633779065944213245, 20963601300674244910397534828794, 344117353602393170461608383214200982125

Offset: 0

N. J. A. Sloane, Feb 16 2004

T = 1
- (1/2 + 1/4 + 1/6 + ... + 1/(2m_1))
+ (1/3 + 1/5 + 1/7 + ... + 1/(2m_2+1))
- (1/(2m_1+2) + 1/(2m_1+4) + ... + 1/(2m_3)
+ (1/(2m_2+3) + 1/(2m_2+5) + ... + 1/(2m_4+1))
- (1/(2m_3+2) + 1/(2m_3+4) + ... + 1/(2m_5)
+ (1/(2m_4+3) + 1/(2m_4+5) + ... + 1/(2m_6+1))
- ...
where the partial sums of the terms from 1 through the end of rows 0, 1, ... are respectively 1, just < -2, just > 3, just < -4, just > 5, etc.
Every positive number appears exactly once as a denominator in T.
The series T is a divergent rearrangement of the conditionally convergent series Sum_{ j>=1} (-1)^j/j which has the entire real number system as its set of limit points.

			1 - (1/2 + 1/4 + 1/6 + ... + 1/454) = -2.002183354..., which is just less than -2; so a(1) = 2m_1 = 454.
1 - (1/2 + 1/4 + 1/6 + ... + 1/454) + (1/3 + 1/5 + ... + 1/45891) = 3.000021113057..., which is just greater than 3; so a(1) = 2m_2 + 1 = 45891.

B. R. Gelbaum and J. M. H. Olmsted, Counterexamples in Analysis, Holden-Day, San Francisco, 1964; see p. 55.

Cf. A092324 (essentially the same), A002387, A056053, A092318, A092317, A092315.

Cf. A092273.

a(2) and a(3) from Hugo Pfoertner, Feb 17 2004

a(4) onwards from Hans Havermann, Feb 18 2004

A092324 Values m_0 = 0, m_1, m_2, ... associated with divergent series T shown below.

Original entry on oeis.org

0, 227, 22945, 273604248, 1506633655224, 980847385203985367, 294892816889532972106622, 10481800650337122455198767414397, 172058676801196585230804191607100491062

Offset: 0

N. J. A. Sloane, Feb 16 2004

T = 1
- (1/2 + 1/4 + 1/6 + ... + 1/(2m_1))
+ (1/3 + 1/5 + 1/7 + ... + 1/(2m_2+1))
- (1/(2m_1+2) + 1/(2m_1+4) + ... + 1/(2m_3)
+ (1/(2m_2+3) + 1/(2m_2+5) + ... + 1/(2m_4+1))
- (1/(2m_3+2) + 1/(2m_3+4) + ... + 1/(2m_5)
+ (1/(2m_4+3) + 1/(2m_4+5) + ... + 1/(2m_6+1))
- ...
where the partial sums of the terms from 1 through the end of rows 0, 1, ... are respectively 1, just < -2, just > 3, just < -4, just > 5, etc.
Every positive number appears exactly once as a denominator in T.
The series T is a divergent rearrangement of the conditionally convergent series Sum_{j>=1} (-1)^j/j which has the entire real number system as its set of limit points.
Comment from Hans Havermann: I calculated these with Mathematica. I used NSum[1/(2i), {i, 1, x}] for the even denominators, where I had to adjust the options to obtain maximal accuracy and N[(EulerGamma + Log[4] - 2)/2 + PolyGamma[0, 3/2 + y]/2, precision] for the odd denominators. The precision needed for the last term shown was around 45 digits.

			1 - (1/2 + 1/4 + 1/6 + ... + 1/454) = -2.002183354..., which is just less than -2; so a(1) = m_1 = 227.
1 - (1/2 + 1/4 + 1/6 + ... + 1/454) + (1/3 + 1/5 + ... + 1/45891) = 3.000021113057..., which is just greater than 3; so a(2) = m_2 = 22945.

B. R. Gelbaum and J. M. H. Olmsted, Counterexamples in Analysis, Holden-Day, San Francisco, 1964; see p. 55.

Cf. A092267 (essentially the same), A002387, A056053, A092318, A092317, A092315.

Cf. A092273.

a(2) and a(3) from Hugo Pfoertner, Feb 17 2004

a(4) onwards from Hans Havermann, Feb 18 2004

A338857 With S(n,k) = Sum_{n<=j<=k} 1/(2*j+1), a(n)=k+1 such that S(n,k-1) < 1 <= S(n,k) for n>=0 and a(0)=1.

Original entry on oeis.org

1, 8, 15, 23, 30, 38, 45, 52, 60, 67, 74, 82, 89, 97, 104, 111, 119, 126, 134, 141, 148, 156, 163, 170, 178, 185, 193, 200, 207, 215, 222, 230, 237, 244, 252, 259, 267, 274, 281, 289, 296, 303, 311, 318, 326, 333, 340, 348, 355, 363, 370, 377, 385, 392, 400, 407, 414

Offset: 0

Gerhard Kirchner, Nov 12 2020

nonn

The following version of the well-known "camel and banana (dates)" problem, is an application of the sequence above:
A camel is to bring full water bags from oasis A to oasis B. It can carry the driver, one full and one empty bag. A full bag is just enough to supply the camel with water for one way from A to B.  What is the minimum reserve a(n) of full bags at oasis A if n full bags are to be delivered at B and depots may be installed along the way?
For details, see link "Transport problem".
n=0: The camel carries one bag which is full in A and empty in B.

			n=0: S(0,k-1)=1 for k=1.
  Thus a(0) = 1+0 = 1.
n=1: S(1,6)=1/3+1/5+...+1/11+1/13=0.995<1, S(1,7)=S(1,6)+1/15=1.022>1.
  Thus a(1) = 7+1 = 8.
n=2: S(2,13)=1/5+1/7+...+1/25+1/27=0.968<1, S(2,14)=S(2,13)+1/29=1.003>1.
  Thus a(2) = 14+1 = 15.

Gerhard Kirchner, Transport problem

Cf. A056053, A092315, A092318, A281355.

Block[{S}, S[n_, k_] := Sum[1/(2 j + 1), {j, n, k}]; {1}~Join~Array[Block[{k = 1}, While[Nand[S[#, k - 1] < 1 <= S[#, k]], k++]; k + 1] &, 56]] (* Michael De Vlieger, Nov 12 2020 *)

block(su: 0, k: 0, n: 1, nmax: 80,
    /*program returns the first nmax terms*/
    v: makelist(0, i, 0, nmax), v[1]: 1,
    while n<=nmax do
    (k: k+1, su: su+1/(2*k+1),
    if su>1 then
     (v[n+1]: k+1, su: su-1/(2*n+1), n: n+1)),
  return(v));

Conjecture: a(n) = ceiling(n*exp(2)+(exp(2)+exp(-2))/(24*n)), verified for n<=3000.

Showing 1-10 of 10 results.

cp's OEIS Frontend

A092315 a(n) is the smallest m such that the partial sum of the odd harmonic series Sum_{j=0..m} 1/(2j+1) is > n.

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A056054 a(n) = smallest even number 2m such that value of odd harmonic series Sum_{j=0..m} 1/(2j) is > n.

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A281355 a(n) = A092318(n) + 1: Number of terms of the odd harmonic series 1 + 1/3 + 1/5 + 1/7 + ... required to reach a sum >= n.

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A091463 a(n) is the smallest j such that 1/1 + 1/4 + 1/7 + ... + 1/j exceeds n.

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A091464 a(n) is the smallest j such that 1/2 + 1/5 + 1/8 + ... + 1/j exceeds n.

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A092317 a(n) = smallest odd number 2m+1 such that the partial sum Sum_{j=0..m} 1/(2j+1) of the odd harmonic series is >= n.

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A091462 a(n) is the smallest j such that 1/3 + 1/6 + 1/9 + ... + 1/j exceeds n.

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A092267 Values 2m_0+1 = 1, 2m_1, 2m_2+1, ... associated with divergent series T shown below.

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A092324 Values m_0 = 0, m_1, m_2, ... associated with divergent series T shown below.

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