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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A074496 a(n) = smallest prime > e^n.

Original entry on oeis.org

2, 3, 11, 23, 59, 149, 409, 1097, 2999, 8111, 22027, 59879, 162779, 442439, 1202609, 3269029, 8886113, 24154957, 65660003, 178482319, 485165237, 1318815761, 3584912873, 9744803489, 26489122147, 72004899361, 195729609461, 532048240609, 1446257064299, 3931334297161
Offset: 0

Views

Author

Joseph L. Pe, Sep 26 2002

Keywords

Examples

			The first prime > e^3 = 20.085... is 23, so a(3) = 23.

Links

Crossrefs

Cf. A000040, A000149, A001113, A014210, A014211, A074497.

Programs

  • Mathematica
    a[n_] := NextPrime[Exp[n]]; a /@ Range[0, 20] (* Giovanni Resta, Apr 03 2017 *)
  • PARI
    for(n=1,50,print1(nextprime(exp(n))","))

Formula

Limmit_{n -> infinity} a(n+1)/a(n) = e. - Jonathan Vos Post, Apr 30 2006

Extensions

More terms from Ralf Stephan, Mar 25 2003
Edited by N. J. A. Sloane, Dec 22 2006
a(18) inserted and more terms added by Amiram Eldar, Sep 30 2019

A091560 Fractional part of e^a(n) is the largest yet.

Original entry on oeis.org

1, 8, 19, 76, 166, 178, 209, 1907, 20926, 22925, 32653, 119136
Offset: 1

Views

Author

Jon Perry, Mar 04 2004

Keywords

Comments

Recursive definition: a(1)=1, a(n) = least number m>a(n-1) such that the fractional part of e^m is greater than the fractional part of e^k for all k, 1<=k
The next such number must be greater than 100000. [Hieronymus Fischer, Jan 06 2009]
a(13) > 300,000. Robert Price, Mar 23 2019

Examples

			a(2)=8, since fract(e^8)= 0.9579870417..., but fract(e^k)<=0.7182818... for 1<=k<=7;
thus fract(e^8)>fract(e^k) for 1<=k<8 and 8 is the minimal exponent > 1 with this property. [_Hieronymus Fischer_, Jan 06 2009]

Crossrefs

Cf. A153663, A153671, A153679, A153687, A153695, A153707, A154130, A153711, A153719, A000149. [Hieronymus Fischer, Jan 06 2009]

Programs

  • Mathematica
    a = 0; Do[b = N[ FractionalPart[ N[ E, 2^12]^n], 24]; If[b > a, Print[n]; a = b], {n, 1, 9400}] (* Robert G. Wilson v, Mar 16 2004 *)
  • PARI
    E=exp(1); /* use sufficient precision! */
ym=0;for(i=1,1000,x=E^i;y=x-floor(x);if(y>ym,print1(","i);ym=y))

Formula

Recursion: a(1):=1, a(k):=min{ m>1 | fract(e^m) > fract(e^a(k-1))}, where fract(x) = x-floor(x). [Hieronymus Fischer, Jan 06 2009]

Extensions

a(8) from Robert G. Wilson v, Mar 16 2004
a(9)-a(11) from Hieronymus Fischer, Jan 06 2009
a(12) from Robert Price, Mar 23 2019

A153707 Greatest number m such that the fractional part of e^A091560(m) >= 1-(1/m).

Original entry on oeis.org

3, 23, 27, 41, 59, 261, 348, 2720, 3198, 6064, 72944, 347065
Offset: 1

Views

Author

Hieronymus Fischer, Jan 06 2009

Keywords

Examples

			a(2)=23, since 1-(1/24) = 0.9583...> fract(e^A091560(2)) = fract(e^8) = 0.95798.. >= 0.95652... >= 1-(1/23).

Crossrefs

Cf. A153663, A153671, A153679, A153687, A153695, A154130, A153715, A153723.
Cf. A000149.

Programs

  • Mathematica
    $MaxExtraPrecision = 100000;
A091560 = {1,8,19,76,166,178,209,1907,20926,22925,32653,119136};
Floor[1/(1-FractionalPart[E^A091560])] (* Robert Price, Apr 18 2019 *)

Formula

a(n):=floor(1/(1-fract(e^A091560(n)))), where fract(x) = x-floor(x).

Extensions

a(12) from Robert Price, Apr 18 2019

A153705 Greatest number m such that the fractional part of e^A153701(n) <= 1/m.

Original entry on oeis.org

1, 2, 11, 11, 23, 28, 69, 85, 115, 964, 1153, 1292, 1296, 1877, 34015, 156075, 952945
Offset: 1

Views

Author

Hieronymus Fischer, Jan 06 2009

Keywords

Examples

			a(3)=11 since 1/12 < fract(e^A153701(3)) = fract(e^3) = 0.0855... <= 1/11.

Crossrefs

Cf. A081464, A153669, A153677, A153685, A153693, A153701, A154130, A153713, A153721.
Cf. A000149.

Programs

  • Mathematica
    A153701 = {1, 2, 3, 9, 29, 45, 75, 135, 219, 732, 1351, 3315, 4795,
   4920, 5469, 28414, 37373};
Table[fp = FractionalPart[E^A153701[[n]]]; m = Floor[1/fp];
While[fp <= 1/m, m++]; m - 1, {n, 1, Length[A153701]}] (* Robert Price, Mar 25 2019 *)

Formula

a(n) = floor(1/fract(e^A153701(n))), where fract(x) = x-floor(x).

A153708 Greatest number m such that the fractional part of e^A153704(n) >= 1-(1/m).

Original entry on oeis.org

3, 23, 27, 261, 348, 2720, 72944, 347065, 244543
Offset: 1

Views

Author

Hieronymus Fischer, Jan 06 2009

Keywords

Examples

			a(2) = 23, since 1-(1/24) = 0.9583... > fract(e^A153704(2)) = fract(e^8) = 0.95798... >= 0.95652... >= 1-(1/23).

Crossrefs

Cf. A153664, A153672, A153680, A153688, A153696, A153704, A154130, A153716, A153724.
Cf. A000149.

Programs

  • Mathematica
    A153704 = {1, 8, 19, 178, 209, 1907, 32653, 119136, 220010};
Table[fp = FractionalPart[E^A153704[[n]]]; m = Floor[1/fp];
While[fp >= 1 - (1/m), m++]; m - 1, {n, 1, Length[A153704]}] (* Robert Price, May 10 2019 *)

Formula

a(n) = floor(1/(1-fract(e^A153704(n)))), where fract(x) = x-floor(x).

Extensions

a(8)-a(9) from Robert Price, May 10 2019

A050809 Primes of the form floor( exp(k) ).

Original entry on oeis.org

2, 7, 65659969, 5184705528587072464087, 14302079958348104463583671072905261080748384225250684971, 17199742630376622641833783925547830057256484050709158699244513
Offset: 1

Views

Author

Patrick De Geest, Oct 15 1999

Keywords

Examples

			a(3) = floor(e^18) = 65659969, which is prime.

Links

Crossrefs

Cf. A050808 (values of k), A000149, A040016, A037028, A000227, A004791.

Programs

  • Mathematica
    Select[Table[Floor[Exp[n]], {n, 150}], PrimeQ] (* Jayanta Basu, Jun 01 2013 *)

Extensions

Corrected by Naohiro Nomoto, Feb 22 2001

A153706 Greatest number m such that the fractional part of e^A153702(n) <= 1/m.

Original entry on oeis.org

1, 2, 11, 11, 964, 34015, 156075, 952945, 170942, 247768, 397506
Offset: 1

Views

Author

Hieronymus Fischer, Jan 06 2009

Keywords

Examples

			a(3) = 11 since 1/12 < fract(e^A153702(3)) = fract(e^3) = 0.0855... <= 1/11.

Crossrefs

Cf. A153662, A153670, A153678, A153686, A153694, A153702, A154130, A153714, A153722.
Cf. A000149.

Programs

  • Mathematica
    Floor[1/(#-Floor[#])]&/@Exp[Select[Range[1000],FractionalPart[E^#]<(1/#)&]] (* Julien Kluge, Sep 20 2016 *)

Formula

a(n) = floor(1/fract(e^A153702(n))), where fract(x) = x - floor(x).

Extensions

a(10)-a(11) from Jinyuan Wang, Mar 03 2020

A001674 a(n) = floor(sqrt( 2*Pi )^n).

Original entry on oeis.org

1, 2, 6, 15, 39, 98, 248, 621, 1558, 3906, 9792, 24546, 61528, 154230, 386597, 969056, 2429063, 6088760, 15262258, 38256809, 95895600, 240374623, 602529828, 1510318305, 3785806567, 9489609784, 23786924200, 59624976768, 149457652641, 374634777972
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A001674 (ceiling sqrt(2 Pi)^n), A017910 (floor sqrt(2)^n), A000149 (floor e^n), A001672 (floor Pi^n), A062541 (floor (Pi*e)^n), A121831 (floor (Pi+e)^n), A032739 (floor (Pi/e)^n), A014217 (floor ((1+sqrt(5))/2)^n).

Programs

  • Mathematica
    Table[Floor[Sqrt[2*Pi]^n], {n, 0, 50}] (* T. D. Noe, Aug 09 2012 *)
  • PARI
    a(n)=(2*Pi)^(n/2)\1 \\ M. F. Hasler, May 29 2018

Extensions

Edited by M. F. Hasler, May 29 2018

A040016 Largest prime < e^n.

Original entry on oeis.org

2, 7, 19, 53, 139, 401, 1093, 2971, 8101, 22013, 59863, 162751, 442399, 1202603, 3269011, 8886109, 24154939, 65659969, 178482289, 485165141, 1318815713, 3584912833, 9744803443, 26489122081, 72004899319, 195729609407, 532048240573, 1446257064289, 3931334297131
Offset: 1

Views

Author

Jud McCranie

Keywords

Comments

A050809 is a subset. Lim_{n --> infinity} a(n+1)/a(n) = e. - Jonathan Vos Post, May 02 2006

Examples

			a(20) = floor(e^20) - 54 = 485165195 - 54 = 485165141 as there are no primes p such that 485165141 < p < 485165195.

Links

Crossrefs

Cf. A000040, A000149, A007512, A014210, A050808, A050809, A059303, A064118, A095935, A115019, A074496, A118840.

Programs

Extensions

Edited by N. J. A. Sloane, Dec 22 2006
a(27)-a(29) from Giovanni Resta, Apr 29 2017

A072630 Values of n where A072629 switches from 01010.. into 0000.. or back.

Original entry on oeis.org

1, 7, 19, 53, 147, 403, 1095, 2979, 8103, 22025, 59873, 162753, 442413, 1202603, 3269017, 8886109, 24154951, 65659969, 178482299, 485165195, 1318815733, 3584912845, 9744803445, 26489122129, 72004899337, 195729609427
Offset: 1

Views

Author

Labos Elemer, Jun 28 2002

Keywords

Crossrefs

See also A004648, A072608, A072609, A072610, A072630.

Programs

  • Mathematica
    m[x_] := Mod[x*Floor[Log[x]//N],2]; Do[s=m[n]+m[n+1]; s1=m[n+1]+m[n+2]; If[ !Equal[s1,s],Print[n]],{n,1,1000000}]

Formula

See program below.
a(n) = A000149(n) or A000149(n)-1 whichever is odd. [From Max Alekseyev, Feb 06 2010]

Extensions

More terms from Max Alekseyev, Feb 06 2010
Previous Showing 11-20 of 43 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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