A000149 -id:A000149 - cp's OEIS frontend

A051102 Floor of exp(n-th prime).

7, 20, 148, 1096, 59874, 442413, 24154952, 178482300, 9744803446, 3931334297144, 29048849665247, 11719142372802611, 639843493530054949, 4727839468229346561, 258131288619006739623, 104137594330290877971834, 42012104037905142549565934, 310429793570191990870734214

Offset: 1

Joel Patrick Hollins (s1161557(AT)cedarville.edu)

nonn

			e = 2.718281828..., e^5 = 148.4131591..., floor( e^5 ) = 148.

Kumanduri and Romero, "Number Theory", Upper Saddle River, NJ, 1998.

T. D. Noe, Table of n, a(n) for n=1..100

Cf. A000040, A000149, A001113.

Floor[Exp[#]]&/@Prime[Range[20]] (* Harvey P. Dale, Dec 12 2012 *)

from sympy import floor, E, prime
def a(n):  return floor(E**prime(n))
print([a(n) for n in range(1, 19)]) # Michael S. Branicky, Jul 20 2021

a(n) = A000149(A000040(n)). - Alois P. Heinz, Apr 09 2020

A055739 [e^n]-th prime.

Original entry on oeis.org

2, 3, 17, 71, 251, 857, 2767, 8803, 27211, 82939, 249779, 744949, 2201273, 6463081, 18858529, 54764947, 158330573, 456016933, 1309050653, 3746543923, 10694444393, 30453898201, 86534078387, 245401348403, 694683409429, 1963275871663, 5540095680547, 15611517864749

Offset: 0

Robert G. Wilson v, Jun 09 2000

Amiram Eldar, Table of n, a(n) for n = 0..39

Cf. A000149, A033844, A006988.

Table[Prime[Floor[N[E^n]]],{n,0,25}] (* Typographical error corrected by Harvey P. Dale, Dec 27 2019 *)

a(n) = prime(A000149(n)). - Amiram Eldar, Jul 22 2025

a(25)-a(27) from Amiram Eldar, Jul 22 2025

A248873 a(n) = floor(e^(n+1) - e^n).

Original entry on oeis.org

1, 4, 12, 34, 93, 255, 693, 1884, 5122, 13923, 37847, 102880, 279658, 760190, 2066413, 5617093, 15268842, 41505016, 112822331, 306682894, 833650539, 2266097111, 6159890600, 16744318683, 45515777207, 123724710091, 336318631172, 914208823689, 2485077232852, 6755140284380

Offset: 0

Danny Rorabaugh, Mar 04 2015

e^(n+1)-e^n-1 < a(n) <= A064780(n) <= a(n)+1 < e^(n+1)-e^n+1.

Lim_{n->infinity} a(n)/e^(n+1) = (e-1)/e. [Corrected by Altug Alkan, Apr 25 2018]

Danny Rorabaugh, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, e
Eric Weisstein's World of Mathematics, Floor Function

Cf. A000149, A001113, A064780, A068996.

Floor[#[[2]]-#[[1]]]&/@Partition[E^Range[0,30],2,1] (* Harvey P. Dale, Jul 24 2018 *)

vector(30, n, n--; floor(exp(n+1)-exp(n))) \\ Michel Marcus, Mar 06 2015

a(n) = floor(e^(n+1)-e^n).

A309087 a(n) = Sum_{k >= 0} floor(n^k / k!).

Original entry on oeis.org

1, 2, 6, 18, 50, 143, 397, 1088, 2973, 8093, 22014, 59861, 162742, 442396, 1202589, 3268996, 8886090, 24154933, 65659949, 178482278, 485165168, 1318815708, 3584912818, 9744803414, 26489122097, 72004899306, 195729609397, 532048240570, 1446257064252

Offset: 0

Rémy Sigrist, Jul 11 2019

nonn

This sequence is inspired by the Maclaurin series for the exponential function.

The series in the name is well defined; for any n > 0, only the first A065027(n) terms are different from zero.

			For n = 3:
- we have:
  k  floor(3^k / k!)
  -  ---------------
  0                1
  1                3
  2                4
  3                4
  4                3
  5                2
  6                1
  >=7              0
- hence a(3) = 1 + 3 + 4 + 4 + 3 + 2 + 1 = 18.

Wikipedia, Taylor series: Exponential function

See A309103, A309104, A309105 for similar sequences.

Cf. A000149, A065027.

a(n) = { my (v=0, d=1); for (k=1, oo, if (d<1, return (v), v += floor(d); d *= n/k)) }

a(n) ~ exp(n) as n tends to infinity.
a(n) <= A000149(n).
a(n) = A309104(n) + A309105(n).

A003619 Not of form [ e^m ], m >= 1.

Original entry on oeis.org

1, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53

Offset: 1

N. J. A. Sloane, Mira Bernstein

nonn

If 1 is excluded (of form [e^0]) then complement of A000149. - Michel Marcus, Jun 16 2013

J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 11.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J. Lambek and L. Moser, Inverse and complementary sequences of natural numbers, Amer. Math. Monthly, 61 (1954), 454-458.

Cf. A000195.

a003619 n = n + floor (log (x + fromIntegral (floor $ log x)))
            where x = fromIntegral n + 1
-- Reinhard Zumkeller, Mar 17 2015

Table[n + Floor@ Log[n + 1 + Floor@ Log[n + 1]], {n, 50}] (* Michael De Vlieger, Oct 06 2017 *)

a(n) = n + floor( log (n + 1 + floor( log (n + 1) )) ) \\ Michel Marcus, Jun 16 2013

a(n) = n + [ log (n + 1 + [ log (n + 1) ]) ].

A061293 a(n) = floor( n^e ), e = 2.718281828...

Original entry on oeis.org

1, 6, 19, 43, 79, 130, 198, 285, 392, 522, 677, 858, 1066, 1304, 1573, 1875, 2211, 2583, 2992, 3440, 3927, 4457, 5029, 5646, 6309, 7019, 7777, 8585, 9445, 10356, 11322, 12343, 13419, 14554, 15747, 17000, 18315, 19692, 21133, 22638, 24210

Offset: 1

Amarnath Murthy, Apr 26 2001

A000290(n) <= a(n) <= A000578(n). - Reinhard Zumkeller, Mar 17 2015

			a(5) = floor(5^e) = floor(79.4323591662132397382254690058565...) = 79.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A000149.

Cf. A000290, A000578.

a061293 = floor . (** exp 1) . fromIntegral
-- Reinhard Zumkeller, Mar 17 2015

for n from 1 to 150 do printf("%d,",floor(n^exp(1))) od;

a[n_]:=Floor[n^E]; (* Vladimir Joseph Stephan Orlovsky, Dec 12 2008 *)

More terms from Winston C. Yang (winston(AT)cs.wisc.edu), May 19 2001

A062277 a(n) = floor(e^n / n^e).

Original entry on oeis.org

2, 1, 1, 1, 1, 3, 5, 10, 20, 42, 88, 189, 414, 921, 2077, 4737, 10921, 25416, 59646, 141033, 335752, 804258, 1937372, 4690989, 11412140, 27884328, 68407056, 168446547, 416226830, 1031816793, 2565591729, 6397371713, 15994440540

Offset: 1

Henry Bottomley, Jul 02 2001

nonn

e is the only positive real k for which k^n is greater than or equal to n^k for all positive real n.

			a(1) = floor(e^1 / 1^e) = floor(e) = 2.

Harry J. Smith, Table of n, a(n) for n = 1..200

Cf. A000149, A061293.

Array[Floor[E^#/#^E] &, 33] (* Michael De Vlieger, Jul 01 2018 *)

{ default(realprecision, 100); e=exp(1); for (n=1, 200, write("b062277.txt", n, " ", floor(e^n / n^e)) ) } \\ Harry J. Smith, Aug 03 2009

A085421 a(0)=2, a(1)=1, a(n+2)=floor[(e-1/e)*a(n+1)+a(n-2)].

Original entry on oeis.org

2, 1, 4, 10, 27, 73, 198, 538, 1462, 3974, 10802, 29363, 79816, 216962, 589764, 1603144, 4357797, 11845720, 32200005, 87528688, 237927642, 646754385, 1758060692, 4778904432, 12990409077, 35311592938, 95986861417

Offset: 0

Gary W. Adamson, Jun 29 2003

nonn

a(n+1)/a(n) converges to e.
For n>0, floor[log a(n)] = n-1.
This resembles a Lucas sequence.

Cf. A000149, A085422, A085839.

Edited by Don Reble, Nov 14 2005

A105461 The floor(exp(n))-th irregular prime.

Original entry on oeis.org

37, 59, 149, 389, 809, 2543, 8089, 25301, 76493, 232439, 695407, 2040551, 5993249, 17532407, 50970511, 147468721, 424835869, 1219642051

Offset: 1

Robert G. Wilson v, Apr 07 2005

Cf. A000928, A000149, A055739, A105457, A105466.

ip={ the list of irregular primes to 12 million }; Table[ ip[[ Floor[E^n]]], {n, 0, 12}]

a(n) = A000928(A000149(n)). - Amiram Eldar, Mar 05 2019

a(13) corrected and a(14)-a(18) added by Amiram Eldar, Mar 05 2019

A116472 a(n) = floor(exp(2*n)).

Original entry on oeis.org

1, 7, 54, 403, 2980, 22026, 162754, 1202604, 8886110, 65659969, 485165195, 3584912846, 26489122129, 195729609428, 1446257064291, 10686474581524, 78962960182680, 583461742527454, 4311231547115195, 31855931757113756, 235385266837019985, 1739274941520501047

Offset: 0

Milton L. Brown (miltbrown(AT)earthlink.net), Jan 20 2008

nonn

A bisection of A000149, which is the main entry for this sequence.

a(n) = A000149(2*n).

Georg Fischer, Table of n, a(n) for n = 0..100 (first 36 terms from Vincenzo Librandi)

Cf. A000149.

a116472 = a000149 . (* 2)  -- Reinhard Zumkeller, Mar 17 2015

[Floor(Exp(2*n)): n in [0..25]]; // Vincenzo Librandi, Aug 23 2011

Floor[Exp[2Range[0,30]]] (* Harvey P. Dale, Aug 22 2011 *)

a(n)=exp(2*n)\1 \\ Charles R Greathouse IV, Aug 23 2011

More terms from Harvey P. Dale, Aug 22 2011

cp's OEIS Frontend

A051102 Floor of exp(n-th prime).

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A055739 [e^n]-th prime.

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A248873 a(n) = floor(e^(n+1) - e^n).

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A309087 a(n) = Sum_{k >= 0} floor(n^k / k!).

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A003619 Not of form [ e^m ], m >= 1.

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A061293 a(n) = floor( n^e ), e = 2.718281828...

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A062277 a(n) = floor(e^n / n^e).

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