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

Author

N. J. A. Sloane, Feb 16 2004

Keywords

• nonn

Comments

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), 0

The 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

Links

• 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

Crossrefs

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

Cf. A074599, A025547, A281355.

Programs

Mathematica

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

Maxima

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 */

Formula

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

Extensions

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

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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