A000799 -id:A000799 - cp's OEIS frontend

A129797 a(n) = floor(7^n/n).

7, 24, 114, 600, 3361, 19608, 117649, 720600, 4483734, 28247524, 179756976, 1153440600, 7453000800, 48444505203, 316504100662, 2077058160600, 13684147881600, 90467422106136, 599941851861744, 3989613314880600

Offset: 1

Mohammad K. Azarian, May 18 2007

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..300

Cf. A000799, A092763, A053638.

[Floor(7^n / n): n in [1..25]]; // Vincenzo Librandi, May 08 2011

Table[Floor[7^n/n],{n,30}] (* Harvey P. Dale, Dec 05 2012 *)

A129798 a(n) = floor(8^n/n).

Original entry on oeis.org

8, 32, 170, 1024, 6553, 43690, 299593, 2097152, 14913080, 107374182, 780903144, 5726623061, 42288908760, 314146179364, 2345624805922, 17592186044416, 132458812569720, 1000799917193443, 7585009898729256

Offset: 1

Mohammad K. Azarian, May 18 2007

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..300

Cf. A000799, A092763, A053638.

[Floor(8^n / n): n in [1..25]]; // Vincenzo Librandi, May 08 2011

Table[Floor[8^n/n],{n,20}] (* Harvey P. Dale, Aug 03 2019 *)

A129799 a(n) = floor(9^n/n).

Original entry on oeis.org

9, 40, 243, 1640, 11809, 88573, 683281, 5380840, 43046721, 348678440, 2852823600, 23535794706, 195528140640, 1634056603925, 13726075472976, 115813761803240, 981010688215680, 8338590849833284, 71097458824894320

Offset: 1

Mohammad K. Azarian, May 18 2007

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..300

Cf. A000799, A092763, A053638.

[Floor(9^n / n): n in [1..25]]; // Vincenzo Librandi, May 08 2011

A128092 a(n) = largest multiple of n which is <= 2^n.

Original entry on oeis.org

2, 4, 6, 16, 30, 60, 126, 256, 504, 1020, 2046, 4092, 8190, 16380, 32760, 65536, 131070, 262134, 524286, 1048560, 2097144, 4194300, 8388606, 16777200, 33554425, 67108860, 134217702, 268435440, 536870910, 1073741820, 2147483646

Offset: 1

Leroy Quet, Feb 14 2007

nonn

Cf. A128093, A000799.

Cf. A000079, A015910.

a:=n->n*floor(2^n/n): seq(a(n),n=1..37); # Emeric Deutsch, Feb 16 2007

f[n_] := n*Floor[2^n/n];Array[f, 33] (* Ray Chandler, Feb 19 2007 *)

def A128092(n): return (m:=1<Chai Wah Wu, Aug 24 2023

a(n) = n*floor(2^n/n) = n*A000799(n).

a(n) = 2^n - (2^n mod n). - Chai Wah Wu, Aug 24 2023

Extended by Emeric Deutsch and Ray Chandler, Feb 19 2007

A000801 a(n) = Sum_{k = 1..n} floor(2^k / k).

Original entry on oeis.org

2, 4, 6, 10, 16, 26, 44, 76, 132, 234, 420, 761, 1391, 2561, 4745, 8841, 16551, 31114, 58708, 111136, 211000, 401650, 766372, 1465422, 2807599, 5388709, 10359735, 19946715, 38459505, 74250899, 143524565, 277742293, 538043341, 1043333611, 2025040421

Offset: 1

N. J. A. Sloane

nonn

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 = 1..500

Cf. A000799.

A000801 := proc(n) local k; sum(floor(2^k/k),k=1..n); end;

Table[Sum[Floor[2^k/k], {k, n}], {n, 30}] (* T. D. Noe, Jun 20 2012 *)

A065482 a(n) = round( 2^n/n ).

Original entry on oeis.org

2, 2, 3, 4, 6, 11, 18, 32, 57, 102, 186, 341, 630, 1170, 2185, 4096, 7710, 14564, 27594, 52429, 99864, 190650, 364722, 699051, 1342177, 2581110, 4971027, 9586981, 18512790, 35791394, 69273666, 134217728, 260301048, 505290270, 981706811, 1908874354, 3714566310

Offset: 1

N. J. A. Sloane, Dec 03 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..300

Cf. A000799, A053638, A082893-A082900.

[Round(2^n/n): n in [1..30]]; // G. C. Greubel, Jan 18 2018

Table[Floor[(Floor[n/2]+2^n)/n], {n, 1, 100}]

a(n) = { round(2^n/n) } \\ Harry J. Smith, Oct 20 2009

def A065482(n): return ((1<>1))//n # Chai Wah Wu, Apr 23 2025

a(n) = A082894(n)/n.

a(n) = floor((2^n + floor(n/2))/n).

A071354 Floor(2^n/n) is odd.

Original entry on oeis.org

12, 18, 25, 36, 42, 45, 48, 55, 80, 91, 95, 98, 99, 100, 108, 110, 112, 125, 130, 132, 135, 136, 140, 143, 152, 153, 155, 160, 161, 162, 175, 184, 187, 190, 192, 198, 208, 216, 224, 225, 228, 232, 235, 238, 240, 242, 245, 247, 248, 261, 266, 273, 275, 279, 285, 286, 289

Offset: 1

R. K. Guy, Jun 12 2002

A student asked if the floor of 2^n / n was always even. He had a proof when n is prime. There is a shorter proof if you look at the binomial expansion of (1+1)^p.

There are infinitely many numbers in this sequence. (Because if n is even, then 2^n*12-n-2 is even, so 2^(2^n*12-n-2) is 4 (mod 6). Define x so that this is 6*x + 4, then dividing by 3 gives 2*x + (4/3), and the floor is an odd number.) - Jinyuan Wang, Oct 13 2018

Cf. A000799, A071941.

Select[ Range[300], OddQ[ Floor[2^# / # ]] & ]

for(n=1,1000,if((-1)^(floor(2^n/n))==-1+isprime(n),print1(n,",")))

from itertools import count, islice
from sympy import isprime
def A071354_gen(startvalue=1): # generator of terms >= startvalue
    yield from filter(lambda k:not isprime(k) and (1<A071354_list = list(islice(A071354_gen(),20)) # Chai Wah Wu, Apr 23 2025

More terms from several correspondents, Jun 12 2002

A215894 a(n) = floor(2^n / n^k), where k is the largest integer such that 2^n >= n^k.

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 4, 6, 1, 1, 2, 3, 5, 9, 1, 1, 2, 4, 6, 10, 17, 1, 2, 3, 5, 9, 15, 26, 1, 2, 4, 6, 11, 18, 31, 1, 2, 4, 6, 11, 19, 32, 1, 2, 3, 5, 9, 16, 28, 49, 1, 2, 4, 7, 13, 22, 38, 1, 1, 3, 5, 9, 16, 27, 47, 1, 2, 3, 5, 10, 17, 30, 51, 1, 2, 3, 5, 10

Offset: 2

Alex Ratushnyak, Aug 25 2012

nonn

a(n) < n.

n such that a(n) = n-1: 2, 3, 996, 3389, 149462.

			a(2) = floor(2^2 / 2^2) = 1,
a(3) = floor(2^3 / 3) = 2,
a(4)..a(9) are floor(2^n / n^2),
a(10)..a(15) are floor(2^n / n^3),
a(16)..a(22) are floor(2^n / n^4), and so on.

Cf. A215892, A000799, A060505.

[Floor(2^n div n^Floor(n *Log(n,2))): n in [2..100]]; // Vincenzo Librandi, Jan 09 2019

Table[Floor[2^n/n^Floor[n Log[n, 2]]], {n, 2, 64}] (* Alonso del Arte, Aug 26 2012 *)

import math
def modiv(a,b):
    return a - b*(a//b)
def modlg(a,b):
    return a // b**int(math.log(a,b))
for n in range(2,100):
    a = 2**n
    print(modlg(a,n), end=',')

a(n) = modlg(2^n, n) = floor(2^n / n^floor(n*logn(2))), where logn is the logarithm base n.

In the base-b representation of k, modlg(k,b) is the most significant digit: k = c0 + c1*b + c2*b^2 + ... + cn*b^n, cn = modlg(k,b), c0 = k mod b. - Alex Ratushnyak, Aug 30 2012

A141602 Integer part of 2^n/log(2^n).

Original entry on oeis.org

2, 2, 3, 5, 9, 15, 26, 46, 82, 147, 268, 492, 909, 1688, 3151, 5909, 11123, 21010, 39809, 75638, 144073, 275050, 526182, 1008516, 1936352, 3723754, 7171675, 13831089, 26708310, 51636066, 99940774, 193635250, 375535031, 728979766, 1416303547

Offset: 1

Cino Hilliard, Aug 21 2008

nonn

2^n/log(2^n) is an approximation to the number of primes < 2^n.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000799, A007525, A050500, A185192.

A141602:= func< n | Floor(2^n/(n*Log(2))) >;
[A141602(n): n in [1..40]]; // G. C. Greubel, Sep 21 2024

Floor[2^#/Log[2^#]]&/@Range[40] (* Harvey P. Dale, Mar 11 2013 *)

g(n) = for(x=1,n,y=floor(2^x/log(2^x));print1(y","))

a(n) = 2^n\log(2^n); \\ Michel Marcus, Feb 24 2021

def A141602(n): return int(2^n/(n*log(2)))
[A141602(n) for n in range(1,41)] # G. C. Greubel, Sep 21 2024

a(n) = A050500(2^n) = floor(2^n*A007525/n) >= A000799(n). - R. J. Mathar, Jan 05 2009

A364881 First significant digit of the decimal expansion of n/(2^n).

Original entry on oeis.org

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

Offset: 1

Ejder Aysun, Aug 10 2023

a(n) is also the first digit of n*5^n = A036291(n).

			n     n/(2^n)
1     0.5                            a(1) = 5
2     0.5                            a(2) = 5
3     0.375                          a(3) = 3
4     0.25                           a(4) = 2
5     0.15625                        a(5) = 1
6     0.9375                         a(6) = 9
7     0.0546875                      a(7) = 5
8     0.03125                        a(8) = 3
9     0.017578125                    a(9) = 1
10    0.009765625                    a(10) = 9
...

Cf. A000030, A000079, A000799, A036291, A111395.

a:= n-> parse((""||(n*5^n))[1]):
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 18 2023

Table[Floor[n/(2^n)/10^Floor[Log10[n/(2^n)]]], {n, 100000}]

def A364881(n): return (n*5**(m:=len(str((1<>n-m) % 10 # Chai Wah Wu, Aug 24 2023

a(n) = floor(n/(2^n)/10^floor(log_10(n/(2^n)))), for n > 0.
a(n) = floor(n/A000079(n)/10^floor(log_10(n/A000079(n)))).
a(n) = floor(A036291(n)/10^floor(log_10(A036291(n)))).
a(n) = A000030(A036291(n)).

cp's OEIS Frontend

A129797 a(n) = floor(7^n/n).

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Magma

Mathematica

A129798 a(n) = floor(8^n/n).

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Magma

Mathematica

A129799 a(n) = floor(9^n/n).

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Magma

A128092 a(n) = largest multiple of n which is <= 2^n.

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Maple

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A000801 a(n) = Sum_{k = 1..n} floor(2^k / k).

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Maple

Mathematica

A065482 a(n) = round( 2^n/n ).

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Magma

Mathematica

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A071354 Floor(2^n/n) is odd.

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Mathematica

PARI

Python

Extensions

A215894 a(n) = floor(2^n / n^k), where k is the largest integer such that 2^n >= n^k.

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Magma