A060505 -id:A060505 - cp's OEIS frontend

A062278 a(n) = floor(3^n / n^3).

3, 1, 1, 1, 1, 3, 6, 12, 27, 59, 133, 307, 725, 1743, 4251, 10509, 26285, 66430, 169450, 435848, 1129505, 2947131, 7737583, 20430377, 54226471, 144621405, 387420489, 1042127936, 2813988985, 7625597484, 20733556989, 56549688380

Offset: 1

Henry Bottomley, Jul 02 2001

nonn

3 is the only integer value of k for which floor(n^k / k^n) is always positive. For positive real x and k, the only value of k for which x^k is always greater than or equal to k^x is e = 2.71828...

			a(2) = floor(3^2 / 2^3) = floor(9/8) = 1.

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

Cf. A000244, A000578, A024026, A060505.

List([1..35],n->Int(3^n/n^3)); # Muniru A Asiru, Jul 01 2018

seq(floor(3^n/n^3),n=1..35); # Muniru A Asiru, Jul 01 2018

Table[Floor[3^n/n^3],{n,40}] (* Vladimir Joseph Stephan Orlovsky, Apr 12 2011 *)

{ default(realprecision, 50); for (n=1, 200, write("b062278.txt", n, " ", floor(3^n / n^3)) ) } \\ Harry J. Smith, Aug 03 2009

A233441 a(n) = floor(2^n / n^3).

Original entry on oeis.org

2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 9, 16, 26, 44, 76, 131, 226, 393, 689, 1213, 2147, 3818, 6818, 12228, 22012, 39768, 72084, 131072, 239027, 437102, 801393, 1472896, 2713342, 5009438, 9267786, 17179869, 31906432, 59362467, 110632938, 206519839, 386111079

Offset: 1

Alex Ratushnyak, Dec 09 2013

nonn

Paolo Xausa, Table of n, a(n) for n = 1..1000

Cf. A000079, A000578, A060505, A062278.

Cf. A230664, A233442.

Array[Floor[2^#/#^3] &, 50] (* Paolo Xausa, Mar 27 2025 *)

for n in range(1,100):  print(2**n // n**3, end=', ')

a(n) = floor(A000079(n) / A000578(n)).

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

A230664 a(n) = floor(3^n/n^2).

Original entry on oeis.org

3, 2, 3, 5, 9, 20, 44, 102, 243, 590, 1464, 3690, 9433, 24402, 63772, 168151, 446851, 1195742, 3219560, 8716961, 23719621, 64836900, 177964421, 490329056, 1355661775, 3760156550, 10460353203, 29179582212, 81605680576, 228767924549, 642740266684, 1809590028175

Offset: 1

Alex Ratushnyak, Dec 11 2013

nonn

Cf. A000244, A000290, A060505, A062278.

Cf. A233441, A233471.

Table[Floor[3^n/n^2], {n, 40}] (* Wesley Ivan Hurt, Apr 25 2023 *)

for n in range(1,100):  print(3**n // n**2, end=", ")

a(n) = floor(A000244(n)/A000290(n)).

A062283 Square array read by descending antidiagonals: T(n,k) = floor(n^k/k^n).

Original entry on oeis.org

1, 0, 2, 0, 1, 3, 0, 0, 1, 4, 0, 1, 1, 1, 5, 0, 1, 1, 0, 0, 6, 0, 1, 1, 1, 0, 0, 7, 0, 2, 3, 1, 0, 0, 0, 8, 0, 4, 6, 3, 1, 0, 0, 0, 9, 0, 6, 12, 6, 2, 0, 0, 0, 0, 10, 0, 10, 27, 16, 4, 1, 0, 0, 0, 0, 11, 0, 16, 59, 39, 11, 2, 0, 0, 0, 0, 0, 12, 0, 28, 133, 104, 33, 6, 1, 0, 0, 0, 0, 0, 13

Offset: 1

Henry Bottomley, Jul 02 2001

			T(3,2) = floor(3^2/2^3) = floor(9/8) = 1.

Paolo Xausa, Table of n, a(n) for n = 1..11325 (first 150 antidiagonals).

Cf. A055651, A060505.

Table[Floor[k^(n-k+1)/(n-k+1)^k], {n, 15}, {k, n}] (* Paolo Xausa, May 06 2024 *)

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A062278 a(n) = floor(3^n / n^3).

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A233441 a(n) = floor(2^n / n^3).

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

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A230664 a(n) = floor(3^n/n^2).

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A062283 Square array read by descending antidiagonals: T(n,k) = floor(n^k/k^n).

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Mathematica