A000149 -id:A000149 - cp's OEIS frontend

A117869 Partial sums of A000149.

1, 3, 10, 30, 84, 232, 635, 1731, 4711, 12814, 34840, 94714, 257468, 699881, 1902485, 5171502, 14057612, 38212564, 103872533, 282354833, 767520028, 2086335762, 5671248608, 15416052054, 41905174183, 113910073520, 309639682948

Offset: 0

Jonathan Vos Post, May 02 2006

Cf. A000149.

Accumulate[Floor[E^Range[0,4000]]] (* Harvey P. Dale, Jul 18 2011 *)

A119604 Merged values of A014217 = {floor(((1+sqrt(5))/2)^n)}, A000149 = {floor(e^n)}, and A001672 = {floor(Pi^n)}, with multiplicity.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 4, 6, 7, 9, 11, 17, 20, 29, 31, 46, 54, 76, 97, 122, 148, 199, 306, 321, 403, 521, 842, 961, 1096, 1364, 2206, 2980, 3020, 3571, 5777, 8103, 9349, 9488, 15126, 22026, 24476

Offset: 0

Bryan David Levenson, Jun 03 2006

			a(8)=6 because floor(((1+sqrt(5))/2)^4)=6, a(9)=7 because floor(e^2)=7 and a(10)=9 because floor(Pi^2)=9.

OEIS index entries related to powers of irrational constants.

Cf. A001672, A014217, A000149.

Edited by M. F. Hasler, May 29 2018

A000227 Nearest integer to e^n.

Original entry on oeis.org

1, 3, 7, 20, 55, 148, 403, 1097, 2981, 8103, 22026, 59874, 162755, 442413, 1202604, 3269017, 8886111, 24154953, 65659969, 178482301, 485165195, 1318815734, 3584912846, 9744803446, 26489122130, 72004899337, 195729609429, 532048240602, 1446257064291, 3931334297144, 10686474581524

Offset: 0

N. J. A. Sloane

x = e^n is the location of the maximum of x^(1/x^(1/n)). One can define another sequence, c(n) as the value of the natural number k that maximizes k^(1/k^(1/n)). Empirically, despite the rounding, c(n) and a(n) match each other until at least n>24500 (see the link). - Stanislav Sykora, Jun 06 2014

Federal Works Agency, Work Projects Administration for the City of NY, Tables of the Exponential Function. National Bureau of Standards, Washington, DC, 1939.
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 230.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..200
Stanislav Sykora, Comments on A000227

Cf. A000149 (floor e^n), A001671 (e^n rounded up), A002160 (nearest integer to Pi^n).

Digits := 40: [seq(round(exp(n)), n=0..30)];

Table[ Round[ N[E^n] ], {n, 0, 30} ]
Round[E^Range[0,30]] (* Harvey P. Dale, Jul 28 2025 *)

apply( A000227(n)=exp(n)\/1, [0..50]) \\ An error message will say so if default(realprecision) must be increased. - M. F. Hasler, May 27 2018

A079490 Exp(n) is closer to an integer than any previous exp(k) for 1 <= k < n.

Original entry on oeis.org

1, 3, 8, 19, 45, 75, 135, 178, 209, 732, 1351, 1907, 5469, 28414, 37373, 404055, 902497

Offset: 1

Donald S. McDonald, Jan 20 2003

			a(2) = 3: exp(3) = 20.08... is closer to an integer than exp(1) = 2.718...
At 37373 the difference from an integer is 0.0000010493779591646530966...

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 178, p. 56, Ellipses, Paris 2008.

Cf. A000149, A001671, A004790, A080053.

a = 1; Do[ d = Abs[ Round[E^n] - N[E^n, Ceiling[ Log[10, E^n] + 10]]]; If[d < a, Print[n]; a = d], {n, 1, 50000}]

{default(realprecision,1000); d(x)=abs(x-round(x))}; a(n)=local(m); if(n<2,n>0,n=a(n-1); m=d(exp(n)); until(d(exp(n))

d(x)=x=frac(x); min(x,1-x)
D(n)=localbitprec(n/log(2)+99); d(exp(n))
r=1; for(n=1,4e4, t=D(n); if(tCharles R Greathouse IV, Oct 31 2022

Corrected and extended to 1351 by several correspondents, Jan 20 2003
a(12)-a(15) from Robert G. Wilson v, Jan 20 2003
a(16)-a(17) from Charles R Greathouse IV, Nov 01 2022

A001671 Powers of e rounded up.

Original entry on oeis.org

1, 3, 8, 21, 55, 149, 404, 1097, 2981, 8104, 22027, 59875, 162755, 442414, 1202605, 3269018, 8886111, 24154953, 65659970, 178482301, 485165196, 1318815735, 3584912847, 9744803447, 26489122130, 72004899338, 195729609429, 532048240602, 1446257064292, 3931334297145, 10686474581525

Offset: 0

N. J. A. Sloane

nonn

T. D. Noe, Table of n, a(n) for n = 0..500

Cf. A000149 (floor(e^n)), A000227 (round(e^n)).

Table[Ceiling[E^n], {n, 0, 50}] (* T. D. Noe, Aug 09 2012 *)

apply( A001671(n)=exp(n)\1+!!n, [0..30]) \\ An error message will say so if default(realprecision) must be increased, for large n. - M. F. Hasler, May 27 2018

A153701 Minimal exponents m such that the fractional part of e^m obtains a minimum (when starting with m=1).

Original entry on oeis.org

1, 2, 3, 9, 29, 45, 75, 135, 219, 732, 1351, 3315, 4795, 4920, 5469, 28414, 37373

Offset: 1

Hieronymus Fischer, Jan 06 2009

Recursive definition: a(1)=1, a(n) = least number m>a(n-1) such that the fractional part of e^m is less than the fractional part of e^k for all k, 1<=k

The next such number must be greater than 100000.

a(18) > 300,000. Robert Price, Mar 23 2019

			a(4)=9, since fract(e^9)=0.08392..., but fract(e^k)>=0.08553... for 1<=k<=8; thus fract(e^9)<fract(e^k) for 1<=k<9.

Cf. A153661, A153669, A153677, A153685, A153693, A153705, A154130, A137994, A153717, A000149.

$MaxExtraPrecision = 100000;
p = 1; Select[Range[1, 300000],
If[FractionalPart[E^#] < p, p = FractionalPart[E^#]; True] &] (* Robert Price, Mar 23 2019 *)

Recursion: a(1):=1, a(k):=min{ m>1 | fract(e^m) < fract(e^a(k-1))}, where fract(x) = x-floor(x).

A153702 Numbers k such that the fractional part of e^k is less than 1/k.

Original entry on oeis.org

1, 2, 3, 9, 732, 5469, 28414, 37373, 93638, 136986, 192897

Offset: 1

Hieronymus Fischer, Jan 06 2009

Numbers k such that fract(e^k) < 1/k, where fract(x) = x-floor(x).
The next such number must be greater than 100000.
a(12) > 300000. - Robert Price, Mar 23 2019

			a(4) = 9 since fract(e^9) = 0.08392... < 1/9, but fract(e^k) = 0.598..., 0.413..., 0.428..., 0.633..., 0.957... for 4 <= k <= 8, which are all greater than 1/k.

Cf. A153662, A153670, A153678, A153686, A153694, A153706, A154130, A153710, A153718.

Cf. A000149.

Select[Range[1000], FractionalPart[E^#] < (1/#) &] (* G. C. Greubel, Aug 24 2016 *)

a(10)-a(11) from Robert Price, Mar 23 2019

A153704 Numbers k such that the fractional part of e^k is greater than 1-(1/k).

Original entry on oeis.org

1, 8, 19, 178, 209, 1907, 32653, 119136, 220010

Offset: 1

Hieronymus Fischer, Jan 06 2009

Numbers k such that fract(e^k) > 1-(1/k), where fract(x) = x-floor(x).
The next such number must be greater than 100000.
a(10) > 300,000. Robert Price, Mar 23 2019

			a(2)=8, since fract(e^8) = 0.957987... >0.875 = 1-(1/8), but fract(e^k) = 0.389..., 0.085..., 0.598..., 0.413..., 0.428..., 0.633... for 2<=k<=7 which all are less than 1-(1/k).

Cf. A153664, A153672, A153680, A153688, A153696, A153708, A154130, A153712, A153720.

Cf. A000149.

Select[Range[2000], N[FractionalPart[E^#], 1000] >= 1 - (1/#) &] (* G. C. Greubel, Aug 25 2016 *)

a(8)-a(9) from Robert Price, Mar 23 2019

A040014 Number of primes < e^n.

Original entry on oeis.org

0, 1, 4, 8, 16, 34, 79, 183, 429, 1019, 2466, 6048, 14912, 37128, 93117, 234855, 595341, 1516233, 3877186, 9950346, 25614562, 66124777, 171141897, 443963543, 1154106844, 3005936865, 7842921261, 20496470801, 53645077679, 140599114669, 368973074565, 969455391690, 2550043255883

Offset: 0

Jud McCranie

nonn

a(n) = A000720(A000149(n)). - Reinhard Zumkeller, Mar 17 2015

David Baugh, Table of n, a(n) for n = 0..59 (terms n = 0..39 from Giovanni Resta, terms n = 40..59 found using Kim Walisch's primecount program).
Index entries for sequences related to numbers of primes in various ranges

Cf. A006880 and A007053.

Cf. A000720, A000149.

a040014 = a000720 . a000149  -- Reinhard Zumkeller, Mar 17 2015

Mathematica
```
Table[PrimePi[Exp[n]], {n, 0, 33}]
```

a(27)-a(29) from Robert G. Wilson v, Jun 09 2000

a(30)-a(32) from Seiichi Manyama, May 04 2016

A050808 Numbers k such that floor(exp(k)) is prime.

Original entry on oeis.org

1, 2, 18, 50, 127, 141, 267, 310, 2290, 4487, 5391, 14025

Offset: 1

Patrick De Geest, Oct 15 1999

Eric Weisstein's World of Mathematics, e-Prime

Cf. A050809 (the actual primes), A000149, A040016, A037028, A000227, A004791, A059791, A059792.

Do[ If[ PrimeQ[ Floor[ \[ExponentialE]^n] ], Print[n] ], {n, 0, 4750} ]
Select[Range[15000],PrimeQ[Floor[Exp[#]]]&] (* Harvey P. Dale, Oct 16 2012 *)

is(n)=ispseudoprime(exp(n)\1) \\ Charles R Greathouse IV, Jan 03 2014

Corrected by Naohiro Nomoto, Feb 22 2001
More terms from Vladeta Jovovic, Feb 24 2001
More terms from Robert G. Wilson v, May 09 2001
a(11) = 5391 from Eric W. Weisstein, May 01 2006
a(12) from Donovan Johnson, Feb 04 2008

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cp's OEIS Frontend

A117869 Partial sums of A000149.

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A119604 Merged values of A014217 = {floor(((1+sqrt(5))/2)^n)}, A000149 = {floor(e^n)}, and A001672 = {floor(Pi^n)}, with multiplicity.

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A000227 Nearest integer to e^n.

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A079490 Exp(n) is closer to an integer than any previous exp(k) for 1 <= k < n.

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A001671 Powers of e rounded up.

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A153701 Minimal exponents m such that the fractional part of e^m obtains a minimum (when starting with m=1).

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A153702 Numbers k such that the fractional part of e^k is less than 1/k.

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A153704 Numbers k such that the fractional part of e^k is greater than 1-(1/k).

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A040014 Number of primes < e^n.