cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A067922 Engel expansion of log(5).

Original entry on oeis.org

1, 2, 5, 11, 27, 33, 103, 110, 133, 193, 999, 1921, 5277, 39085, 279568, 365167, 870685, 1539231, 3794071244, 9761250028, 10906977906, 18144658921, 103136388637, 134389561244, 139712478096, 667384153244, 731220192045, 1317705035571, 1664555384503, 3008856758047
Offset: 1

Views

Author

Benoit Cloitre, Mar 03 2002

Keywords

Crossrefs

See A006784 for explanation of Engel expansions.
Cf. A016628.

Programs

  • PARI
    \\ a(1)=1 and for n>1:
s=log(5); for(i=1,30,s=s*ceil(1/s)-1; print1(ceil(1/s),","); );

A067923 Engel expansion of log(23).

Original entry on oeis.org

1, 1, 1, 8, 12, 135, 199, 378, 600, 1836, 4897, 8198, 8993, 84887, 450287, 892157, 5053447, 5183243, 15350505, 19963471, 31631271, 37655416, 2138752269, 4805947342, 14508700588, 27508373127, 28635924075, 30814114095, 32073629885, 961160400603, 3607716972786
Offset: 1

Views

Author

Benoit Cloitre, Mar 03 2002

Keywords

Crossrefs

See A006784 for explanation of Engel expansions. Log(23) is the first number of the form Log(n), n an integer, for which it is not known whether a BBP formula exists.
Cf. A016646.

Programs

  • PARI
    \\ a(1)=1 and for n>1:
s=log(23); for(i=1,30,s=s*ceil(1/s)-1; print1(ceil(1/s),","); );

A068375 Engel expansion of exp(2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 6, 333, 444, 476, 735, 1227, 1740, 16541, 19348, 45751, 49117, 139069, 567376, 1438204, 23992692, 36889249, 69357744, 206213866, 487816917, 23325191834, 82451809814, 378427041146, 431057010568, 793887257125
Offset: 1

Views

Author

Benoit Cloitre, Mar 03 2002

Keywords

Comments

Cf. A006784

A068376 Engel expansion of log(1+e).

Original entry on oeis.org

1, 4, 4, 83, 87, 430, 660, 9039, 47028, 128352, 157861, 241465, 362709, 9031153, 11990180, 89103542, 102393518, 318327464, 361501552, 1119756916, 1310133018, 1995850459, 20709593111, 94478870265, 105076191441, 123229448284, 5619694006942, 8033666701549
Offset: 1

Views

Author

Benoit Cloitre, Mar 03 2002

Keywords

Comments

Cf. A006784

A068478 Engel expansion of Gamma(3/4) = Sum_{i>0} 1/Product_{n=1..i} 1/a(n).

Original entry on oeis.org

1, 5, 8, 60, 11640, 14881, 23098, 90040, 239883, 1631365, 2803282, 4276595, 4343763, 10834865, 50457394, 101387685, 349770616, 384832508, 1025023303, 1582124479, 7753484647, 14554741880, 110768341257, 246113848342
Offset: 1

Views

Author

Benoit Cloitre, Mar 10 2002

Keywords

Comments

The Gamma function is defined by Gamma(0)=1 and the functional equation Gamma(x+1) = (x+1)*Gamma(x).

Crossrefs

Cf. A006784.

Programs

  • PARI
    \\ 500 significant digits
s=gamma(3/4); for(i=0,40,s=s*ceil(1/s)-1; print1(ceil(1/s), ", "); );

A068479 Engel expansion of Gamma(1/4)=3.62560990822190831193...

Original entry on oeis.org

1, 1, 1, 2, 4, 205, 4009, 5523, 14063, 32669, 47704, 196620, 589791, 782889, 804393, 849445, 976320, 1256587, 1789793, 9067576, 17283444, 18567900, 24793858, 59704212, 60741675, 130071084, 216799081, 1518169585, 4849913556
Offset: 1

Views

Author

Benoit Cloitre, Mar 10 2002

Keywords

Comments

The Gamma function is defined by Gamma(0)=1 and the functional equation Gamma(x+1)=(x+1)*Gamma(x) Gamma(1/4)=3.62560990822190831193...

Crossrefs

Cf. A006784.

Programs

  • PARI
    s=gamma(1/4); for(i=0,40,s=s*ceil(1/s)-1; print1(ceil(1/s),","); );

Formula

Gamma(1/4)=sum(i>0, prod(n=1, i, 1/a(n)))

A071857 Engel expansion of sqrt(2*log(2)).

Original entry on oeis.org

1, 6, 16, 32, 279, 726, 4141, 4368, 54482, 112572, 366613, 978019, 5342223, 41589964, 201780051, 353794663, 408307432, 463394050, 676353989, 866725306, 999357112, 3878963429, 4169753024, 8541140255, 23422387081, 26113359872, 940995107440, 1104573841707
Offset: 1

Views

Author

Benoit Cloitre, Jun 09 2002

Keywords

Comments

a(1)=1, then PARI program gives a(n) for n>1.

Crossrefs

Cf. A064619, A006784.

Programs

  • PARI
    s= sqrt(2*log(2)); for(i=1,30,s=s*ceil(1/s)-1; print1(ceil(1/s),","); );

A076303 Engel expansion of exp(Pi * sqrt(163)) - 262537412640768743.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 19, 1169, 21384, 520409, 2559029, 2922819, 3228884, 6972029, 18244654, 24601850, 146539491, 620041946, 865572355, 1298955860, 3005000777, 5169423076, 6941400197, 9965578146, 26183561695, 39614218376
Offset: 1

Views

Author

Robert G. Wilson v, Mar 03 2003

Keywords

Comments

262537412640768743.9999999999992500... is Ramanujan's constant which is extremely close to an integer. The Engel expansion of the fractional part begins with 40 terms 2.

Links

Crossrefs

Cf. A006784, A060295.

Programs

  • Mathematica
    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[E^(Pi*Sqrt[163]) - 262537412640768743, 52]
  • PARI
    default(realprecision, 100000); r=exp(Pi*sqrt(163))-262537412640768743; for(i=1, 100, s=r*ceil(1/r)-1; print1(ceil(1/r), ", "); r=s); /* Georg Fischer, Nov 21 2020 */

Extensions

More terms from Georg Fischer, Nov 21 2020

A080230 Engel expansion for (positive) constant defined in A078756.

Original entry on oeis.org

2, 4, 30, 63, 162, 629, 3918, 5345, 8577, 10902, 12073, 19001, 28961, 33387, 65944, 75653, 105116, 179007, 262900, 504773, 844426, 1120997, 3272709, 5732988, 11239007, 25703795, 29618846, 40805409, 46574664, 123045405, 163155335, 11300437000, 23399972470
Offset: 1

Views

Author

Jon Perry, Feb 08 2003

Keywords

Crossrefs

Cf. A006784, A078756, A080231.

Programs

  • PARI
    { my(x=vector(15), a=vector(#x-1)); x[1]=0.62923321313860758778445822033246963679442280615921;
for (i=1,#a,a[i]=ceil(1/x[i]); x[i+1]=x[i]*a[i]-1); a }

Formula

See A006784 for definition of Engel expansion.

Extensions

a(15) onward corrected and more terms from Sean A. Irvine, Sep 09 2025

A080231 Engel expansion for (negative) constant defined in A078756.

Original entry on oeis.org

-1, -2, -3, -4, -9, -11, -17, -82, -86, -89, -259, -1031, -1493, -3088, -18107, -3128320, -4245574, -4431505, -7141303, -25396934, -87452973, -220313229, -235578556, -1892660806, -5160802315, -5697316936, -6909396926, -17605431920, -18782195284, -19291817256
Offset: 1

Views

Author

Jon Perry, Feb 08 2003

Keywords

Crossrefs

Cf. A006784, A078756, A080230.

Programs

  • PARI
    { my(x=vector(18), a=vector(#x-1)); x[1]=-0.62923321313860758778445822033246963679442280615921;
for (i=1, #a, a[i]=ceil(1/x[i]); x[i+1]=x[i]*a[i]-1); a }

Formula

See A006784 for definition of Engel expansion

Extensions

a(18) onward corrected by Sean A. Irvine, Sep 09 2025
Previous Showing 91-100 of 108 results. Next

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