A007525 -id:A007525 - cp's OEIS frontend

A165313 Triangle T(n,k) = A091137(k-1) read by rows.

1, 1, 2, 1, 2, 12, 1, 2, 12, 24, 1, 2, 12, 24, 720, 1, 2, 12, 24, 720, 1440, 1, 2, 12, 24, 720, 1440, 60480, 1, 2, 12, 24, 720, 1440, 60480, 120960, 1, 2, 12, 24, 720, 1440, 60480, 120960, 3628800, 1, 2, 12, 24, 720, 1440, 60480, 120960, 3628800, 7257600, 1, 2, 12

Offset: 1

Paul Curtz, Sep 14 2009

From a study of modified initialization formulas in Adams-Bashforth (1855-1883) multisteps method for numerical integration. On p.36, a(i,j) comes from (j!)*a(i,j) = Integral_{u=i,..,i+1} u*(u-1)*...*(u-j+1) du; see p.32.
Then, with i vertical, j horizontal, with unreduced fractions, partial array is:
0) 1,  1/2,  -1/12,   1/24,  -19/720,   27/1440, ... =  1/log(2)
1) 1,  3/2,   5/12,  -1/24,   11/720,  -11/1440, ... =  2/log(2)
2) 1,  5/2,  23/12,   9/24,  -19/720,   11/1440, ... =  4/log(2)
3) 1,  7/2,  53/12,  55/24,  251/720,  -27/1440, ... =  8/log(2)
4) 1,  9/2,  95/12, 161/24, 1901/720,  475/1440, ... = 16/log(2)
5) 1, 11/2, 149/12, 351/24, 6731/720, 4277/1440, ... = 32/log(2)
... [improved by Paul Curtz, Jul 13 2019]
First line: the reduced terms are A002206/A002207, logarithmic or Gregory numbers G(n). The difference between the second line and the first one is 0 together A002206/A002207. This is valid for the next lines. - Paul Curtz, Jul 13 2019
See A141417, A140825, A157982, horizontal numerators: A141047, vertical numerators: A000012, A005408, A140811, A141530, A157411. On p.56, coefficients are s(i,q) = (1/q!)* Integral_{u=-i-1,..,1} u*(u+1)*...*(u+q-1) du.
Unreduced fractions array is:
-1) 1,   1/2,  5/12,   9/24, 251/720, 475/1440, ... = A002657/A091137
0)  2,   0/2,  4/12,   8/24, 232/720, 448/1440, ... = A195287/A091137
1)  3,  -3/2,  9/12,   9/24, 243/720, 459/1440, ...
2)  4,  -8/2, 32/12,   0/24, 224/720, 448/1440, ...
3)  5, -15/2, 85/12, -55/24, 475/720, 475/1440, ...
...
(on p.56 up to 6)). See A147998. Vertical numerators: A000027, A147998, A152064, A157371, A165281.
From Paul Curtz, Jul 14 2019: (Start)
Difference table from the second line and the first one difference:
1,                -1/2,      -1/12,    -1/24,    -19/720, -27/1440, ...
-3/2,             5/12,       1/24,    11/720,    11/1440, ...
23/12,           -9/24,    -19/720,  -11/1440, ...
-55/24,        251/720,    27/1440, ...
1901/720,    -475/1440,
-4277/1440, ...
...
Compare the lines to those of the first array.
The verticals are the signed diagonals of the first array. (End)

			1;
1,2;
1,2,12;
1,2,12,24;
1,2,12,24,720;

P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Centre de Calcul Scientifique de l'Armement, Note 12, Arcueil, 1969.

Cf. A090624, A091137.
Cf. A000012, A000079, A002657, A005408, A007525, A131920, A140811, A140825, A141047, A141417, A141530, A157411, A157982, A195287.
Cf. A002206, A002207.

(* a = A091137 *) a[n_] := a[n] = Product[d, {d, Select[Divisors[n]+1, PrimeQ]}]*a[n-1]; a[0]=1; Table[Table[a[k-1], {k, 1, n}], {n, 1, 11}] // Flatten (* Jean-François Alcover, Dec 18 2014 *)

A059542 Beatty sequence for 1 + 1/log(2).

Original entry on oeis.org

2, 4, 7, 9, 12, 14, 17, 19, 21, 24, 26, 29, 31, 34, 36, 39, 41, 43, 46, 48, 51, 53, 56, 58, 61, 63, 65, 68, 70, 73, 75, 78, 80, 83, 85, 87, 90, 92, 95, 97, 100, 102, 105, 107, 109, 112, 114, 117, 119, 122, 124, 127, 129, 131, 134, 136, 139, 141, 144, 146, 149, 151

Offset: 1

Mitch Harris, Jan 22 2001

Harry J. Smith, Table of n, a(n) for n = 1..2000
Aviezri S. Fraenkel, Jonathan Levitt, Michael Shimshoni, Characterization of the set of values f(n)=[n alpha], n=1,2,..., Discrete Math. 2 (1972), no.4, 335-345.
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Beatty complement is A059541.

Cf. A002162, A007525.

Floor[Range[100]*(1 + 1/Log[2])] (* Paolo Xausa, Jul 05 2024 *)

{ default(realprecision, 100); b=1 + 1/log(2); for (n = 1, 2000, write("b059542.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

a(n) = floor(n*(1+1/log(2))). - Michel Marcus, Jan 04 2015

a(n) = n+A307513(n). - R. J. Mathar, Jan 04 2020

A059183 Engel expansion of 1/log(2) = 1.4427...

Original entry on oeis.org

1, 3, 4, 4, 5, 5, 5, 6, 47, 109, 935, 4763, 7821, 8895, 9889, 35798, 44347, 1146551, 7874944, 8043393, 27403243, 34058912, 58098040, 68760470, 80046897, 560099631, 611427977, 14235032003, 602865059026, 813485869378

Offset: 1

Mitch Harris, May 16 2003

Cf. A006784 for definition of Engel expansion.

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

G. C. Greubel, Table of n, a(n) for n = 1..1000
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.
Index entries for sequences related to Engel expansions

Cf. A007525 (1/log(2)).

EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@
NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]/1} &, {Ceiling[1/(A - Floor[A])], (A - Floor[A])/1}, n - 1]];
EngelExp[N[1/Log[2], 7!], 100] (* Modified by G. C. Greubel, Dec 27 2016 *)

A141602 Integer part of 2^n/log(2^n).

Original entry on oeis.org

2, 2, 3, 5, 9, 15, 26, 46, 82, 147, 268, 492, 909, 1688, 3151, 5909, 11123, 21010, 39809, 75638, 144073, 275050, 526182, 1008516, 1936352, 3723754, 7171675, 13831089, 26708310, 51636066, 99940774, 193635250, 375535031, 728979766, 1416303547

Offset: 1

Cino Hilliard, Aug 21 2008

nonn

2^n/log(2^n) is an approximation to the number of primes < 2^n.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000799, A007525, A050500, A185192.

A141602:= func< n | Floor(2^n/(n*Log(2))) >;
[A141602(n): n in [1..40]]; // G. C. Greubel, Sep 21 2024

Floor[2^#/Log[2^#]]&/@Range[40] (* Harvey P. Dale, Mar 11 2013 *)

g(n) = for(x=1,n,y=floor(2^x/log(2^x));print1(y","))

a(n) = 2^n\log(2^n); \\ Michel Marcus, Feb 24 2021

def A141602(n): return int(2^n/(n*log(2)))
[A141602(n) for n in range(1,41)] # G. C. Greubel, Sep 21 2024

a(n) = A050500(2^n) = floor(2^n*A007525/n) >= A000799(n). - R. J. Mathar, Jan 05 2009

A307513 Beatty sequence for 1/log(2).

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 25, 27, 28, 30, 31, 33, 34, 36, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 51, 53, 54, 56, 57, 59, 60, 62, 63, 64, 66, 67, 69, 70, 72, 73, 75, 76, 77, 79, 80, 82, 83, 85, 86, 88, 89, 90, 92, 93, 95, 96, 98, 99, 100, 102, 103, 105, 106, 108, 109, 111, 112, 113, 115

Offset: 1

R. J. Mathar, Apr 12 2019

Very similar to A059539 because A002581 is close to A007525.

Cf. A007525.

a(n) = floor(n*A007525).

A166986(n) = 2*a(n+2)-4.

A242053 Decimal expansion of 1/log(2)-1, the mean value of a random variable following the Gauss-Kuzmin distribution.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Aug 13 2014

			0.4426950408889634073599246810018921374266459541529859341354494...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.17 Gauss-Kuzmin-Wirsing constant, p. 151.

Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, 2.17 p. 21.
Michael Penn, This infinite series is crazy!, YouTube video, 2020.
Index entries for transcendental numbers

Cf. A007525.

RealDigits[1/Log[2] - 1, 10, 99] // First

Equals (1/log(2))*Integral_{x=0..1} x/(1+x) dx.

Equals Sum_{k>=1} 1/(2^k*(1 + 2^(2^(-k)))). - Amiram Eldar, May 28 2021

A371667 Decimal expansion of -Ei(-1) / log(2).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 10 2024

			0.316504114203126786893754620753862815669085943...

Cf. A007525, A019704, A099285, A232734.

RealDigits[-ExpIntegralEi[-1]/Log[2], 10, 103][[1]]

eint1(1)/log(2) \\ Michel Marcus, Apr 11 2024

Equals Integral_{x=0..oo} exp(-2^x) dx.

cp's OEIS Frontend

A165313 Triangle T(n,k) = A091137(k-1) read by rows.

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A059542 Beatty sequence for 1 + 1/log(2).

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A059183 Engel expansion of 1/log(2) = 1.4427...

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A141602 Integer part of 2^n/log(2^n).

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A307513 Beatty sequence for 1/log(2).

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A242053 Decimal expansion of 1/log(2)-1, the mean value of a random variable following the Gauss-Kuzmin distribution.

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A371667 Decimal expansion of -Ei(-1) / log(2).

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