A006784 -id:A006784 - cp's OEIS frontend

A108905 Engel series expansion for Sierpinski's constant.

1, 1, 2, 6, 51, 114, 136, 149, 1388, 3654, 3961, 41614, 2975365, 4120126, 5760908, 11210809, 21235067, 43239302, 156258546, 230627452, 595763433, 4709180601, 148918280487, 841708059551, 2895551449652, 5660433533409, 6575950336213

Offset: 0

Hauke Worpel (hw1(AT)email.com), Jul 17 2005

nonn

Cf. A062089, A006784.

EngelExp[A_, n_] := Join[ Array[ 1 &, Floor[ A]], First @ Transpose @ NestList[{Ceiling[ 1/Expand[ #[[1]] #[[2]] -1]], Expand[ #[[1]] #[[2]] -1]} &, {Ceiling[ 1/(A - Floor[A]) ], A - Floor[A]}, n - 1]]; EngelExp[ N[ Pi(-Log[Pi] + 2EulerGamma + 4LogGamma[3/4]), 2^8], 25] (* Robert G. Wilson v *)

a(11) - a(26) from Robert G. Wilson v, Jul 21 2005

A130818 Decimal expansion of number whose Engel expansion is the sequence of squares, that is, 1, 4, 9, 16,...

Original entry on oeis.org

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

Offset: 1

Stephen Casey (hexomino(AT)gmail.com), Jul 17 2007

			1.2795853023360672674372044408115333532858411...

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

Stephen Crowley, Two New Zeta Constants: Fractal String, Continued Fraction, and Hypergeometric Aspects of the Riemann Zeta Function, arXiv:1207.1126 [math.NT], 2012, page 17.
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.
Eric Weisstein's World of Mathematics, Engel Expansion
Eric Weisstein's World of Mathematics, Modified Bessel Function of the First Kind

Cf. A006784, A064648, A101689.

RealDigits[BesselI[0, 2] - 1, 10, 105] // First (* Jean-François Alcover, Oct 01 2013 *)

besseli(0,2)-1 \\ Charles R Greathouse IV, Oct 01 2013

Equal to Sum_{n>=1} 1/n!^2 or BesselI(0,2) - 1. - Gerald McGarvey, Nov 12 2007

Equals A070910 - 1. - R. J. Mathar, Jun 13 2008

A161559 Engel expansion of tan(1/2).

Original entry on oeis.org

2, 11, 54, 136, 1473, 3483, 36244, 41086, 305728, 379955, 582669, 4540387, 5020443, 22096761, 24228660, 48364856, 178868536, 234516235, 524137295, 1096104266, 11627672572, 19828856452, 31372689367, 47247829739, 701124643395

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 13 2009

nonn

			A161011 = 0.546302.... = 1/2+1/(2*11)+1/(2*11*54)+1/(2*11*54*136)+ ...

Index to sequences related to Engel expansions

Cf. A006784.

EngelExp[A_,n_]:=Join[Array[1&,Floor[A]],First@Transpose@NestList[{Ceiling[1/ Expand[ #[[1]]#[[2]]-1]],Expand[ #[[1]]#[[2]]-1]}&,{Ceiling[1/(A-Floor[A])], A-Floor[A]},n-1]]; EngelExp[N[Tan[1/2],6! ],50]

Example added by R. J. Mathar, Oct 04 2009

A185443 Engel expansion of A060997 = 1.433127...

Original entry on oeis.org

1, 3, 4, 6, 6, 10, 10, 10, 21, 66, 207, 722, 6563, 25007, 372733, 2028763, 5472218, 41430101, 75142985, 192675195, 201216921, 925285050, 935598827, 2288358581, 2346034092, 26271379744, 41588896504, 152594692251, 529451874660

Offset: 1

Jani Melik, Feb 04 2011

nonn

Eric W. Weisstein, Engel Expansion

Cf. A006784.

Digits := 5000:
a0 := evalf(BesselI(0,2)/BesselI(1,2)):
f1 := proc(n) local i, an, u, a:
an := [ ]:
u := n:
for i from 1 to 30 do
   a := ceil(1/u):
   an := [ op(an), a ]:
   u := u * a - 1:
od:
RETURN (an): end:  f1(a0);

CFB(v)={ \\ converts a continued fraction to a number
my(x=v[#v]*1.);
forstep(i=#v-1,1,-1,
x = v[i] + x^-1;
);
x
};
Engel(x)={
my(v=List(),t);
while(1,
trap(,
return(Vec(v))
,
t = ceil(1/x)
);
listput(v,t);
x = (x * t) - 1
)
};
\p 500
Engel(CFB(vector(500,i,i)))

gp script from Charles R Greathouse IV, Feb 06 2011

A278765 Engel expansion of natural logarithm of golden ratio.

Original entry on oeis.org

3, 3, 4, 4, 4, 6, 15, 48, 118, 147, 671, 1026, 3075, 44641, 48364, 1868380, 75080506, 96848501, 911582093, 2511879981, 8700005050, 15888441652, 108526838262, 446779835336, 632466801279, 1084794852728, 1184722346307, 1657692322844, 12376968750845, 17341469111712, 27996895637798, 38935285631573

Offset: 1

Ilya Gutkovskiy, Nov 28 2016

			log(phi)  = 0.4812118250596... = 1/3 + 1/(3*3) + 1/(3*3*4) + 1/(3*3*4*4) + 1/(3*3*4*4*4) + 1/(3*3*4*4*4*6) + ...

Eric Weisstein's World of Mathematics, Engel Expansion
Eric Weisstein's World of Mathematics, Golden Ratio
Index entries for sequences related to Engel expansions

Cf. A006784 (for definition of Engel expansion).

Cf. A001622, A002390, A028259, A059195.

EngelExp[A_, n_]:=Join[Array[1&, Floor[A]], First@Transpose@NestList[{Ceiling[1/Expand[ #[[1]]#[[2]]-1]], Expand[ #[[1]]#[[2]]-1]}&, {Ceiling[1/(A-Floor[A])], A-Floor[A]}, n-1]]; EngelExp[N[Log[GoldenRatio], 7! ], 40]

A278766 Engel expansion of plastic constant (real root of x^3 - x - 1).

Original entry on oeis.org

1, 4, 4, 6, 6, 27, 74, 86, 372, 853, 947, 1475, 3686, 9084, 19174, 30737, 1530833, 2401466, 2521253, 3649563, 3802245, 9320024, 1092256819, 2114664794, 2878948610, 8842525055, 13769551820, 26996892389, 215947176106, 269439735691, 13694290818678, 18312336654245, 19649485782723, 63266709043539

Offset: 1

Ilya Gutkovskiy, Nov 28 2016

			(1/2+sqrt(23/108))^(1/3) + (1/2-sqrt(23/108))^(1/3) = 1.324717957244... = 1/1 + 1/(1*4) + 1/(1*4*4) + 1/(1*4*4*6) + 1/(1*4*4*6*6) + 1/(1*4*4*6*6*27) + ...

Eric Weisstein's World of Mathematics, Engel Expansion
Eric Weisstein's World of Mathematics, Plastic Constant
Index entries for sequences related to Engel expansions

Cf. A006784 (for definition of Engel expansion).

Cf. A060006, A072117.

EngelExp[A_, n_]:=Join[Array[1&, Floor[A]], First@Transpose@NestList[{Ceiling[1/Expand[ #[[1]]#[[2]]-1]], Expand[ #[[1]]#[[2]]-1]}&, {Ceiling[1/(A-Floor[A])], A-Floor[A]}, n-1]]; EngelExp[N[(1/2 + Sqrt[23/108])^(1/3) + (1/2 - Sqrt[23/108])^(1/3), 7! ], 40]

A279627 Engel expansion of the Glaisher-Kinkelin constant A074962.

Original entry on oeis.org

1, 4, 8, 27, 59, 188, 384, 427, 2525, 71429, 80727, 357492, 13200877, 65161876, 7439912342, 15555881542, 71559279848, 116275866868, 345574982189, 737460049244, 9183275685671, 12641946167319, 126181443702371

Offset: 1

Benedict W. J. Irwin, Dec 16 2016

nonn

See A006784 for more details on the Engel expansion.

			1.2824271291006... = 1/1 + 1/(1*4) + 1/(1*4*8) + 1/(1*4*8*27) + ...

Cf. A074962.

EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]} &, {Ceiling[1/(A - Floor[A])], A - Floor[A]}, n - 1]];
EngelExp[Glaisher, 22]

A346801 Engel expansion of A249270 = lim_{n->oo} (1/n)*Sum_{k=1..n} smallest prime not dividing k.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 13, 18, 160, 162, 1421, 1745, 6097, 6348, 15474, 65948, 103608, 366088, 573005, 1048188, 1138953, 3410520, 6376825, 279823002, 306445433, 1082288597, 1489247033, 2533043524, 5591690612, 8082600305, 22159965172, 62442143259, 70275959860

Offset: 1

Corey Clemons, Aug 04 2021

nonn

Cf. A006784 for definition of Engel expansion.

Eric Weisstein's World of Mathematics, Engel Expansion
Wikipedia, Engel Expansion
Index entries for sequences related to Engel expansions

Cf. A006784, A249270.

More terms from Alois P. Heinz, Aug 04 2021

cp's OEIS Frontend

A108905 Engel series expansion for Sierpinski's constant.

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A130818 Decimal expansion of number whose Engel expansion is the sequence of squares, that is, 1, 4, 9, 16,...

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A161559 Engel expansion of tan(1/2).

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A185443 Engel expansion of A060997 = 1.433127...

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A278765 Engel expansion of natural logarithm of golden ratio.

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A278766 Engel expansion of plastic constant (real root of x^3 - x - 1).

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Mathematica

A279627 Engel expansion of the Glaisher-Kinkelin constant A074962.

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Mathematica

A346801 Engel expansion of A249270 = lim_{n->oo} (1/n)*Sum_{k=1..n} smallest prime not dividing k.

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