A001671 -id:A001671 - cp's OEIS frontend

A103838 Complement of A001671.

2, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 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, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79

Offset: 0

N. J. A. Sloane, Mar 30 2005

nonn

a(n) = n + [log(n + [log(n)])], apart from some initial exceptions. - David W. Wilson, Mar 29 2005

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

Original entry on oeis.org

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

N. J. A. Sloane

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

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..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

Bisection: A116472.
Cf. A001113, A003619, A000079, A000244, A064780 (first differences, apart from initial term).
Cf. A000227 (round e^n), A001671 (ceiling e^n).

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

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

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

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

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

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

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

A059303 Numbers k such that floor(e^k) + 1 is prime.

Original entry on oeis.org

0, 1, 5, 7, 10, 105, 22959, 34888

Offset: 1

Felice Russo, Jan 25 2001

0 followed by all k such that ceiling(e^k) is prime. - Jeppe Stig Nielsen, Feb 12 2024

Eric Weisstein's World of Mathematics, e-Prime

Cf. A001671, A050808.

Select[Range[40000], PrimeQ[Floor[E^#] + 1] &] (* G. C. Greubel, Jan 06 2017 *)

isok(k) = isprime(floor(exp(k))+1) \\ Michel Marcus, Jun 08 2013

Corrected by Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001

a(7)-a(8) from Donovan Johnson

A114650 a(1)=1. For n>1, a(n) is smallest positive integer not among the earlier terms of the sequence such that floor(log(a(n))) does not equal floor(log(a(n-1))).

Original entry on oeis.org

1, 3, 2, 4, 8, 5, 9, 6, 10, 7, 11, 21, 12, 22, 13, 23, 14, 24, 15, 25, 16, 26, 17, 27, 18, 28, 19, 29, 20, 30, 55, 31, 56, 32, 57, 33, 58, 34, 59, 35, 60, 36, 61, 37, 62, 38, 63, 39, 64, 40, 65, 41, 66, 42, 67, 43, 68, 44, 69, 45, 70, 46, 71, 47, 72, 48, 73, 49, 74, 50, 75, 51

Offset: 1

Leroy Quet, Dec 21 2005

Sequence is a permutation of the positive integers. (Sequence A114651 is the inverse permutation.)

Apparently this permutation is completely decomposable into (disjoint) cycles of finite length. The number of fixed points (cf. A114726) seems to be infinite, but for each k>1 there are presumably only finitely many cycles of length k (cf. A114727 and A114728). - Klaus Brockhaus, Dec 29 2005

			Since all positive integers m where floor(log(m)) equals 0 or 1 occur among the first 11 terms of the sequence and since floor(log(a(11))) = 2, then a(12) must be 21 (which is the smallest positive integer m such that floor(log(m)) = 3).

Cf. A114651, A000195, A001671, A114726, A114727, A114728.

More terms from Klaus Brockhaus, Dec 25 2005

A114726 Fixed points of permutation A114650.

Original entry on oeis.org

1, 4, 11, 30, 79, 218, 589, 1604, 4357, 11850, 32203, 87546, 237963, 646864

Offset: 1

Klaus Brockhaus, Dec 29 2005, Jan 10 2006

nonn

Observation: A001671(n) < a(n) < A001671(n+1) for 1 < n <= 14.

Conjecture: Sequence is infinite and lim n -> infinity a(n+1)/a(n) = e = 2.718281828...

			A114650(11) = 11, so 11 is a term.

Cf. A114650, A114727, A001671.

A333534 a(n) is the number of log(n)-smooth numbers <= n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19

Offset: 2

N. J. A. Sloane, Apr 08 2020

nonn

Number of k <= n such that the greatest prime factor of k is <= log(n).

Alois P. Heinz, Table of n, a(n) for n = 2..65536

Cf. A001671, A006530, A051102, A137845, A295084.

A333534 := n -> nops(select(k -> A006530(k) <= ilog(n), [$1..n])):
seq(A333534(n), n=2..86); # Peter Luschny, Apr 09 2020
# second Maple program:
b:= proc(n) option remember; max(1, map(i-> i[1], ifactors(n)[2])) end:
a:= n-> (t-> add(`if`(b(i)<= t, 1, 0), i=1..n))(ilog(n)):
seq(a(n), n=2..100);  # Alois P. Heinz, Apr 09 2020

a[n_] := Select[Range[n], FactorInteger[#][[-1, 1]] <= Log[n]&] // Length;
a /@ Range[2, 100] (* Jean-François Alcover, May 17 2020 *)

gpf(j)={if(j==1,1,my(f=factor(j));f[#f[,2],1])};
for(n=2,80,my(L=log(n));print1(sum(k=1,n,gpf(k)<=L),", ")) \\ Hugo Pfoertner, Apr 09 2020

sm(lim, p)=if(p==2, return(logint(lim\1, 2)+1)); my(s=0, q=precprime(p-1), t=1); for(e=0, logint(lim\=1, p), s+=sm(lim\t, q); t*=p); s
a(n)=if(n<8,return(n>2)); sm(n, precprime(log(n))) \\ Charles R Greathouse IV, Apr 16 2020

a(n) = A096300(n), n>2. - R. J. Mathar, Apr 27 2020

A072632 Solutions to A072631[n]=0.

Original entry on oeis.org

1, 3, 8, 21, 55, 149, 404, 1097, 2981, 8104, 22027, 59875, 162755, 442414, 1202605, 3269018

Offset: 1

Labos Elemer, Jun 28 2002

Essentially the same as A001671.

			Compare with A072610 [related to A004648].

Cf. A004648, A072608, A072609, A072610, A072630, A072631, A072610.

Do[s=Floor[Mod[Floor[n*Log[n]]//N, n]]; If[s==0, Print[n]], {n, 1, 10000000}]

A116938 Expansion of e^2 in base 2.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0

Offset: 3

Jonathan Vos Post, Mar 21 2006

			111.010001000000 (base 2) ~ 7.389056098930650... (base 10) ~ e^2. 100 decimal places precision here.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.
Eli Maor, e: The Story of a Number, Princeton Univ. Press, 1994.

John Cosgrave, links to "New Proofs of the Irrationality of e^2 and e^4."

Cf. A001113 (e), A072334 (e^2), A090142 (e^2-e).
Cf. A090143 (e^3-2e^2+e/2), A089139 (e^4-3e^3+2e^2-e/6), A090143 (e^3-2e^2+e/2).
Cf. A001671 (powers of e rounded up), A107586 (powers of e^(1/e) rounded up).

RealDigits[E^2, 2, 100] (* Stefan Steinerberger, Mar 30 2006 *)

cp's OEIS Frontend

A103838 Complement of A001671.

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

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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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A059303 Numbers k such that floor(e^k) + 1 is prime.

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A114650 a(1)=1. For n>1, a(n) is smallest positive integer not among the earlier terms of the sequence such that floor(log(a(n))) does not equal floor(log(a(n-1))).

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A114726 Fixed points of permutation A114650.

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A333534 a(n) is the number of log(n)-smooth numbers <= n.

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Maple