A000149 a(n) = floor(e^n).

1, 2, 7, 20, 54, 148, 403, 1096, 2980, 8103, 22026, 59874, 162754, 442413, 1202604, 3269017, 8886110, 24154952, 65659969, 178482300, 485165195, 1318815734, 3584912846, 9744803446, 26489122129, 72004899337, 195729609428, 532048240601, 1446257064291

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

A000079(n) <= a(n) <= A000244(n); for n > 0: A064780(n) = a(n+1) - a(n). - Reinhard Zumkeller, Mar 17 2015

Satisfies Benford's law [Whyman et al., 2016]. - N. J. A. Sloane, Feb 12 2017

References

• 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).

Links

• T. D. Noe, Table of n, a(n) for n=0..300
• R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
• Johan Kok, Integer sequences with conjectured relation with certain graph parameters of the family of linear Jaco graphs, arXiv:2507.16500 [math.CO], 2025. See p. 9.
• G. Whyman, N. Ohtori, E. Shulzinger and Ed. Bormashenko, Revisiting the Benford law: When the Benford-like distribution of leading digits in sets of numerical data is expectable?, Physica A: Statistical Mechanics and its Applications, 461 (2016), 595-601.
• Index entries for sequences related to Benford's law

Crossrefs

Bisection: A116472.

Cf. A001113, A003619, A000079, A000244, A064780 (first differences, apart from initial term).

Cf. A000227 (round e^n), A001671 (ceiling e^n).

Programs

Haskell

a000149 = floor . (exp 1 ^)
a000149_list = let e = exp 1 in map floor $ iterate (* e) 1
-- Reinhard Zumkeller, Mar 17 2015

Mathematica

a[n_]:=Floor[E^n]; (* Vladimir Joseph Stephan Orlovsky, Dec 12 2008 *)
Floor[E^Range[0,30]] (* Harvey P. Dale, Apr 01 2012 *)

PARI

a(n) = floor(exp(n)); \\ Arkadiusz Wesolowski, Nov 26 2011

PARI

apply( A000149(n)=exp(n)\1, [0..30]) \\ An error message will say so if default(realprecision) must be increased, for large n. - M. F. Hasler, May 27 2018

Python

from sympy import floor, E
def a(n):  return floor(E**n)
print([a(n) for n in range(29)]) # Michael S. Branicky, Jul 20 2021

Formula

a(n)^(1/n) converges to e because |1-a(n)/e^n|=|e^n-a(n)|/e^n < e^(-n) and so a(n)^(1/n)=(e^n*(1+o(1)))^(1/n)=e*(1+o(1)). - Hieronymus Fischer, Jan 22 2006