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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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%I #52 Dec 10 2022 18:34:43

%S 1,7,56,418,3091,22845,168803,1247297,9216353,68100150,503195828,

%T 3718142207,27473561357,203003686105,1500005624923,11083625711270,

%U 81897532160124,605145459495140,4471453748222756,33039822589391675,244133102611731230,1803913190804074903

%N 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.

%C From _Gerhard Kirchner_, May 21 2020: (Start)

%C 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.

%C 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<t<1. The test can be carried out with t=0. Additionally, the precision can be tested by checking if t=1 makes a difference.

%C 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)

%C 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

%H Gerhard Kirchner, <a href="/A092315/b092315.txt">Table of n, a(n) for n = 1..1000</a>

%H Gerhard Kirchner, <a href="/A092315/a092315.pdf">Asymptotic formulas</a>

%H Grant Sanderson, <a href="https://www.youtube.com/watch?v=851U557j6HE">Researchers thought this was a bug (Borwein integrals)</a>, 3Blue1Brown video (2022).

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Borwein_integral">Borwein integral</a>

%F 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

%F a(n) = floor(exp(2*n-gamma)/4). - _Gerhard Kirchner_, Jul 23 2020

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

%o (Maxima)

%o block(

%o fpprec:1000, gam: %gamma, nmax:1000,

%o fl: openw("bfile1000.txt"),

%o OH(k,t):=(log(4*k)+gam)/2+1/(2*k)-11/(48*k^2)+1/(8*k^3)-127*t/(1920*k^4),

%o printf(fl, "1 1"), newline(fl),

%o for n from 2 thru nmax do

%o (u: bfloat(exp(2*n-gam)/4), k: floor(u),

%o x0: bfloat(OH(k,0)), x01: bfloat(OH(k,1)), x1: bfloat(OH(k-1,0)),

%o n0: floor(x0), n01: floor(x01), n1: floor(x1), m: n,

%o if n0=n and n01=n and n1=n-1 then

%o (h: concat(n, " ", k), printf(fl, h), newline(fl)) else n: nmax),

%o if m<nmax then print(concat("Precision too low: Stop at n= ",m)),

%o close(fl));

%o /* _Gerhard Kirchner_, Jul 23 2020 */

%o /* The first nmax terms are saved as a b-file */

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

%Y Cf. A074599, A025547, A281355.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Feb 16 2004

%E More terms from _M. F. Hasler_, Jan 24 2017

%E a(17) in the data section and 127 terms in the b-file corrected by _Gerhard Kirchner_, Jul 23 2020