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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.

%I A092315 #52 Dec 10 2022 18:34:43
%S A092315 1,7,56,418,3091,22845,168803,1247297,9216353,68100150,503195828,
%T A092315 3718142207,27473561357,203003686105,1500005624923,11083625711270,
%U A092315 81897532160124,605145459495140,4471453748222756,33039822589391675,244133102611731230,1803913190804074903
%N 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.
%C A092315 From _Gerhard Kirchner_, May 21 2020: (Start)
%C A092315 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 A092315 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 A092315 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 A092315 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 A092315 Gerhard Kirchner, <a href="/A092315/b092315.txt">Table of n, a(n) for n = 1..1000</a>
%H A092315 Gerhard Kirchner, <a href="/A092315/a092315.pdf">Asymptotic formulas</a>
%H A092315 Grant Sanderson, <a href="https://www.youtube.com/watch?v=851U557j6HE">Researchers thought this was a bug (Borwein integrals)</a>, 3Blue1Brown video (2022).
%H A092315 Wikipedia, <a href="https://en.wikipedia.org/wiki/Borwein_integral">Borwein integral</a>
%F A092315 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 A092315 a(n) = floor(exp(2*n-gamma)/4). - _Gerhard Kirchner_, Jul 23 2020
%t A092315 A092315[n_] := Floor[Exp[2*n - EulerGamma]/4]; Table[A092315[n], {n, 1, 22}] (* _Robert P. P. McKone_, Jul 13 2021 *)
%o A092315 (Maxima)
%o A092315 block(
%o A092315 fpprec:1000, gam: %gamma, nmax:1000,
%o A092315 fl: openw("bfile1000.txt"),
%o A092315 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 A092315 printf(fl, "1 1"),   newline(fl),
%o A092315 for n from 2 thru nmax do
%o A092315 (u: bfloat(exp(2*n-gam)/4), k: floor(u),
%o A092315 x0: bfloat(OH(k,0)), x01: bfloat(OH(k,1)), x1: bfloat(OH(k-1,0)),
%o A092315 n0: floor(x0), n01: floor(x01), n1: floor(x1),  m: n,
%o A092315 if n0=n and n01=n and n1=n-1 then
%o A092315          (h: concat(n, " ", k), printf(fl, h),  newline(fl)) else n: nmax),
%o A092315 if m<nmax then print(concat("Precision too low: Stop at n= ",m)),
%o A092315 close(fl));
%o A092315 /* _Gerhard Kirchner_, Jul 23 2020 */
%o A092315 /* The first nmax terms are saved as a b-file */
%Y A092315 Except for first term, same as A092318. Equals (A056053-1)/2.
%Y A092315 Cf. A074599, A025547, A281355.
%K A092315 nonn
%O A092315 1,2
%A A092315 _N. J. A. Sloane_, Feb 16 2004
%E A092315 More terms from _M. F. Hasler_, Jan 24 2017
%E A092315 a(17) in the data section and 127 terms in the b-file corrected by _Gerhard Kirchner_, Jul 23 2020