A158468 -id:A158468 - cp's OEIS frontend

A100668 Decimal expansion of the base-2 analog of the Euler-Mascheroni constant.

8, 3, 2, 7, 4, 6, 1, 7, 7, 2, 7, 6, 8, 6, 7, 1, 5, 0, 6, 4, 6, 4, 1, 7, 5, 1, 9, 4, 0, 8, 1, 1, 5, 5, 3, 5, 1, 6, 2, 4, 3, 1, 5, 3, 1, 0, 2, 6, 3, 2, 8, 1, 0, 1, 6, 3, 1, 4, 9, 8, 1, 9, 7, 5, 8, 4, 5, 8, 1, 3, 5, 1, 4, 4, 6, 0, 9, 5, 8, 4, 2, 2, 9, 0, 2, 0, 2, 6, 0, 0, 3, 4, 4, 4, 3, 0, 6, 3, 0, 4, 6, 7, 8, 6, 0

Offset: 0

Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 05 2004

			0.83274617727686715064641751940811553516243153102632810163149819758458\ 1351446095842290202600344430630467860292319092706478030883524064025...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Hartosh Singh Bal and Gaurav Bhatnagar, Prime number conjectures from the Shapiro class structure, arXiv:1903.09619 [math.NT], 2019.
Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant.
Eric Weisstein's World of Mathematics, Harmonic Number (with contribution from Jonathan Sondow).
Eric Weisstein's World of Mathematics, Lg.

Cf. A001620, A002162, A158468.

RealDigits[Limit[Sum[D[Log[2, x], x] /. x -> k, {k, 1, n}] - Log[2, n], n -> Infinity], 10, 135][[1]]

Euler/log(2) \\ Michel Marcus, Jun 02 2020

EulerGamma/log(2) = A001620/A002162.

Equals Integral_{x=-infinity..infinity} x*2^(-x)*log(2)*exp(-2^(-x)) dx. - Alois P. Heinz, Nov 09 2016

A158469 Continued fraction for hz = limit_{k -> infinity} 1 + k - Sum_{j = -k..k} exp(-2^j).

Original entry on oeis.org

1, 3, 189, 3, 2, 2, 1, 5, 4, 1, 1, 3, 1, 1, 1, 5, 8, 12, 1, 22, 7, 14, 1, 2, 1, 5, 1, 4, 222, 1, 1, 2, 3, 24, 6, 27, 1, 15, 1, 9, 1, 1, 18, 6, 24, 2, 1, 7, 1, 4, 2, 2, 1, 1, 84, 1, 1, 1, 3, 1, 1, 1, 1, 1, 5, 15, 3, 13, 3, 2, 14, 1, 1, 1, 10, 15, 10, 1, 6, 120, 1, 31, 2, 4, 2, 7, 2, 2, 1, 1, 1, 1, 1, 3, 7

Offset: 0

Alois P. Heinz, Mar 19 2009

			1.33274738243289922500860109837389970441674398225984453657972 ...

Cf. A158468 (decimal expansion), A159835 (Engel expansion).

with(numtheory): hz:= limit(1+k -sum(exp(-2^j), j=-k..k), k=infinity): cfrac(evalf(hz, 130), 100, 'quotients')[];

terms = 95; digits = terms+15; Clear[f]; f[k_] := f[k] = 1+k-Sum[Exp[-2^j], {j, -k, k}] // RealDigits[#, 10, digits+1]& // First // Quiet; f[1]; f[n = 2]; While[f[n] != f[n-1], n++]; hz = FromDigits[f[n]]*10^-digits; ContinuedFraction[hz, terms] (* Jean-François Alcover, Mar 23 2017 *)

A159835 Engel expansion of hz = limit_{k -> infinity} 1 + k - Sum_{j = -k..k} exp(-2^j).

Original entry on oeis.org

1, 4, 4, 4, 4, 6, 11, 11, 11, 14, 61, 266, 1006, 1030, 1261, 6264, 7583, 7979, 7986, 12386, 80041, 87434, 130927, 270073, 1653819, 1715177, 1973657, 3483485, 12346987, 17531499, 21237674, 84103203, 195088616, 725688944, 2813572082, 3138084145, 10870485195

Offset: 1

Alois P. Heinz, Apr 23 2009

Cf. A006784 for definition of Engel expansion.

			hz = 1.3327473824328992250086010983738997044167439822598445365797 ...

F. Engel, Entwicklung der Zahlen nach Stammbrüchen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmänner in Marburg, 1913, pp. 190-191.

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
Eric Weisstein's World of Mathematics, Engel Expansion
Wikipedia, Engel Expansion
Index entries for sequences related to Engel expansions

Cf. A158468 (decimal expansion), A158469 (continued fraction).

hz:= limit(1+k -sum(exp(-2^j), j=-k..k), k=infinity):
engel:= (r,n)-> `if`(n=0 or r=0, NULL, [ceil(1/r), engel(r*ceil(1/r)-1, n-1)][]):
Digits:=300:
engel(evalf(hz), 39);

Some terms corrected by Alois P. Heinz, Nov 22 2020

A339168 Decimal expansion of Sum_{k>=1} Im(Gamma(1-2kPii/log(2)))/(kPi).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 0, 5, 1, 5, 6, 0, 3, 2, 0, 7, 4, 3, 6, 2, 1, 8, 3, 5, 7, 8, 9, 6, 5, 7, 8, 4, 1, 6, 9, 2, 5, 4, 3, 1, 2, 4, 5, 1, 2, 3, 3, 5, 1, 6, 4, 3, 4, 9, 4, 8, 2, 2, 0, 2, 9, 6, 4, 0, 8, 7, 6, 0, 3, 3, 6, 8, 2, 7, 4, 5, 1, 6, 1, 1, 7, 8, 0, 0, 7, 8, 0, 5, 1, 3, 7, 2, 2, 2, 9, 8, 4, 8, 2, 3, 6, 2, 0, 6, 5, 6

Offset: 0

Alois P. Heinz, Nov 25 2020

			0.00000120515603207436218357896578416925431245123351643494822...

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A001620, A002162, A100668, A158468.

evalf(sum(Im(GAMMA(1-2*k*Pi*I/log(2)))/(k*Pi), k=1..infinity), 120);

Equals A158468 - gamma/log(2) - 1/2.

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A100668 Decimal expansion of the base-2 analog of the Euler-Mascheroni constant.

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A158469 Continued fraction for hz = limit_{k -> infinity} 1 + k - Sum_{j = -k..k} exp(-2^j).

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A159835 Engel expansion of hz = limit_{k -> infinity} 1 + k - Sum_{j = -k..k} exp(-2^j).

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A339168 Decimal expansion of Sum_{k>=1} Im(Gamma(1-2*k*Pi*i/log(2)))/(k*Pi).

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A339168 Decimal expansion of Sum_{k>=1} Im(Gamma(1-2kPii/log(2)))/(kPi).