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A092315
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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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10
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1, 7, 56, 418, 3091, 22845, 168803, 1247297, 9216353, 68100150, 503195828, 3718142207, 27473561357, 203003686105, 1500005624923, 11083625711270, 81897532160124, 605145459495140, 4471453748222756, 33039822589391675, 244133102611731230, 1803913190804074903
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OFFSET
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1,2
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COMMENTS
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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<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.
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
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LINKS
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FORMULA
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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
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MATHEMATICA
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PROG
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(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 m<nmax then print(concat("Precision too low: Stop at n= ", m)),
close(fl));
/* The first nmax terms are saved as a b-file */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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a(17) in the data section and 127 terms in the b-file corrected by Gerhard Kirchner, Jul 23 2020
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STATUS
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approved
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