A000227 -id:A000227 - cp's OEIS frontend

A057219 {e^n}-th prime, where {e^n} is closest integer to e^n, A000227.

2, 5, 17, 71, 257, 857, 2767, 8807, 27239, 82939, 249779, 744949, 2201281, 6463081, 18858529, 54764947, 158330587, 456016949, 1309050653, 3746543951, 10694444393, 30453898201, 86534078387, 245401348403, 694683409451

Offset: 0

Robert G. Wilson v, Sep 16 2000

nonn

Cf. A000227.

Table[ Prime[ Round[ N[ E^n ] ] ], {n, 0, 27} ]

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

A002581 Decimal expansion of cube root of 3.

Original entry on oeis.org

1, 4, 4, 2, 2, 4, 9, 5, 7, 0, 3, 0, 7, 4, 0, 8, 3, 8, 2, 3, 2, 1, 6, 3, 8, 3, 1, 0, 7, 8, 0, 1, 0, 9, 5, 8, 8, 3, 9, 1, 8, 6, 9, 2, 5, 3, 4, 9, 9, 3, 5, 0, 5, 7, 7, 5, 4, 6, 4, 1, 6, 1, 9, 4, 5, 4, 1, 6, 8, 7, 5, 9, 6, 8, 2, 9, 9, 9, 7, 3, 3, 9, 8, 5, 4, 7, 5, 5, 4, 7, 9, 7, 0, 5, 6, 4, 5, 2, 5, 6, 6, 8, 6, 8, 3, 5, 0, 8

Offset: 1

N. J. A. Sloane

The largest k^(1/k), for any natural number k, occurs when k = 3 = A000227(1). - Stanislav Sykora, Jun 04 2014
3^(1/3) is also the Kolmogorov constant C(3,2) in the case supremum norm on the real line. - Jean-François Alcover, Jul 17 2014
(1/3)*log(3) = -lim_{n->oo} (n-th derivative zeta(n+1)) / ((n-1)-th derivative zeta(n)) = 0.3662040962227... Convergence is to 25 digits by n = ~1000. zeta is the Riemann zeta function. - Richard R. Forberg, Feb 24 2015

			1.442249570307408382321638310780109588391869253499350577546416...

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).
Horace S. Uhler, Many-figure approximations for cube root of 2, cube root of 3, cube root of 4 and cube root of 9 with chi_2 data, Scripta Math. 18, (1952), 173-176.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Simon Plouffe, The cube root of 3 to 2000 places. [Wayback Machine link]
Simon Plouffe, The cube root of 3 to 2000 places.
Harry Pollard, Problem E 2190, Elementary Problems, The American Mathematical Monthly, Vol. 76, No. 8 (1969), p. 937; A Special Property of 3, Solutions to Problem E 2190, by Douglas Lind and Charles Wexler, ibid., Vol. 77, No. 7 (1970), p. 768.
Horace S. Uhler, Many-figure approximations for cubed root of 2, cubed root of 3, cubed root of 4, and cubed root of 9 with chi2 data, Scripta Math. 18, (1952). 173-176. [Annotated scanned copies of pages 175 and 176 only]
Eric Weisstein's MathWorld, Landau-Kolmogorov Constants.
Index entries for algebraic numbers, degree 3.

Cf. A002946 (continued fraction).

RealDigits[N[3^(1/3), 200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)

default(realprecision, 20080); x=3^(1/3); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b002581.txt", n, " ", d));  \\ Harry J. Smith, May 07 2009

3^(1/3) >= min(k^(1/m), m^(1/k)) for any positive integers k and m (Pollard, 1969). - Amiram Eldar, Feb 14 2025

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

A002160 Nearest integer to Pi^n.

Original entry on oeis.org

1, 3, 10, 31, 97, 306, 961, 3020, 9489, 29809, 93648, 294204, 924269, 2903677, 9122171, 28658146, 90032221, 282844564, 888582403, 2791563950, 8769956796, 27551631843, 86556004192, 271923706894, 854273519914, 2683779414318, 8431341691876, 26487841119104, 83214007069230

Offset: 0

N. J. A. Sloane

			a(0) = 1 because Pi^0 = 1;
a(2) = 10 because Pi^2 = 9.8696...;
a(10) = 93648 because Pi^10 = 93648.047476...

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. 122.
J. T. Peters, Ten-Place Logarithm Table. Vols. 1 and 2, rev. ed. Ungar, NY, 1957, Vol. 1 (Appendix), p. 1.
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).

Cf. A000227 (e^n), A001672 (floor(Pi^n)), A001673 (ceiling(Pi^n)).

a := []: Digits := 1000: for n from 0 to 50 do: a := [op(a),round(Pi^n)]: od: seq(a[i+1],i=0..50);

Round[Pi^Range[0,40]] (* Harvey P. Dale, Jun 10 2024 *)

apply( A002160(n)=Pi^n\/1, [0..50]) \\ An error message will say so if default(realprecision) must be increased. - M. F. Hasler, May 27 2018

[round(pi^n) for n in range(0,29)] # Stefano Spezia, Jan 15 2025

More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 29 2003

Edited by M. F. Hasler, May 27 2018

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

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

Original entry on oeis.org

2, 7, 65659969, 5184705528587072464087, 14302079958348104463583671072905261080748384225250684971, 17199742630376622641833783925547830057256484050709158699244513

Offset: 1

Patrick De Geest, Oct 15 1999

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

Amiram Eldar, Table of n, a(n) for n = 1..9
Eric Weisstein's World of Mathematics, e-Prime.

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

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

Corrected by Naohiro Nomoto, Feb 22 2001

A001675 a(n) = round(sqrt( 2*Pi )^n).

Original entry on oeis.org

1, 3, 6, 16, 39, 99, 248, 622, 1559, 3907, 9793, 24546, 61529, 154230, 386598, 969056, 2429064, 6088760, 15262259, 38256810, 95895601, 240374624, 602529829, 1510318305, 3785806568, 9489609784, 23786924201, 59624976768, 149457652642, 374634777972

Offset: 0

N. J. A. Sloane

nonn

Cf. A001674 (floor sqrt(2 Pi)^n), A001698 (ceiling sqrt(2 Pi)^n).

Cf. A017911 (round sqrt(2)), A000227 (round e^n), A002160 (round Pi^n).

Table[Floor[Sqrt[2*Pi]^n + 1/2], {n, 0, 50}] (* T. D. Noe, Aug 09 2012 *)
Round[(Sqrt[2*Pi])^Range[0,30] ] (* Harvey P. Dale, Jun 05 2018 *)

apply( a(n)=(2*Pi)^(n/2)\/1, [0..40]) \\ M. F. Hasler, May 29 2018

A243443 Decimal expansion of 7^(1/sqrt(7)).

Original entry on oeis.org

2, 0, 8, 6, 4, 9, 3, 4, 9, 6, 3, 0, 9, 3, 6, 4, 4, 2, 3, 1, 9, 1, 0, 1, 2, 0, 7, 8, 3, 3, 1, 8, 7, 4, 6, 4, 4, 7, 5, 9, 9, 1, 7, 8, 7, 1, 1, 8, 2, 4, 7, 7, 0, 4, 4, 3, 1, 1, 4, 8, 3, 4, 0, 3, 0, 7, 7, 1, 7, 6, 2, 4, 6, 5, 9, 9, 9, 6, 9, 6, 8, 9, 1, 7, 8, 2, 2, 6, 9, 7, 7, 1, 8, 1, 3, 1, 8, 9, 5, 0, 7, 0, 3, 4, 5

Offset: 1

Stanislav Sykora, Jun 05 2014

The largest k^(1/sqrt(k)), for any natural number k, which occurs for k = 7 = A000227(2).

			2.086493496309364423191012078331874644759917871182477044311483403077176...

Stanislav Sykora, Table of n, a(n) for n = 1..2000

Cf. A002581, A243406 (k=8), A243444 (k=6).

RealDigits[7^(1/Sqrt[7]),10,120][[1]] (* Harvey P. Dale, Aug 17 2024 *)

default(realprecision, 20080); 7.0^(1.0/sqrt(7.0));

A291210 Numbers k such that round(kk^(1/k)) - round((k-1)(k-1)^(1/(k-1))) > 1.

Original entry on oeis.org

2, 4, 10, 27, 80, 230, 644, 1780, 4879, 13315, 36261, 98650, 268260, 729326, 1982655, 5389579, 14650584, 39824632, 108254817, 294267376, 799901968, 2174359323, 5910521810, 16066464445, 43673178798, 118716008808, 322703570021, 877199250941

Offset: 1

Hugo Pfoertner, Aug 21 2017

nonn

			Let s(x) = x*x^(1/x); r(x) = round(s(x));
a(1) = 2:
  s(1) = 1,
  s(2) = 2.82842712474619...;
  r(1) = 1,
  r(2) = 3,
  r(2) - r(1) = 2;
a(2) = 4:
  s(3) = 4.32674871...,
  s(4) = 5.6568542...;
  r(3) = 4,
  r(4) = 6,
  r(4) - r(3) = 2;
...
a(19) = 108254817:
  s(108254816) = 108254834.49999999422...,
  s(108254817) = 108254835.50000000346...;
  r(108254816) = 108254834,
  r(108254817) = 108254836,
  r(108254817) - r(108254816) = 2.

Hugo Pfoertner, Table of n, a(n) for n = 1..500

Cf. A000227, A291211, A291212.

f[n_] := Round[n*n^(1/n)]; g[k_] := f[k] > 1 + f[k-1]; A = Select[Range[2, 5000], g]; Do[AppendTo[A, SelectFirst[Floor[E Last@ A] + Range[1000], g]], {n, 19}]; A (* Giovanni Resta, Aug 21 2017 *)

Lim_{n->infinity} a(n)/a(n-1) = e.
It appears that, for most values of n, a(n) = floor(e^(n-1/2) + 7/8) - binomial(n,2). An exception occurs at n = 7; are there more? - Jon E. Schoenfield, Aug 22 2017
No more exceptions found through n = 30000. - Hugo Pfoertner, Aug 25 2017

cp's OEIS Frontend

A057219 {e^n}-th prime, where {e^n} is closest integer to e^n, A000227.

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

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A002581 Decimal expansion of cube root of 3.

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A001671 Powers of e rounded up.

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A002160 Nearest integer to Pi^n.

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

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

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A001675 a(n) = round(sqrt( 2*Pi )^n).

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A291210 Numbers k such that round(kk^(1/k)) - round((k-1)(k-1)^(1/(k-1))) > 1.