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

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

Author

Alex Ratushnyak, Aug 25 2012

Keywords

• nonn

Comments

a(n) < n.

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

Examples

			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.
		

Crossrefs

Cf. A215892, A000799, A060505.

Programs

Magma

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

Mathematica

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

Python

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=',')

Formula

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