A075101 Numerator of 2^n/n.

2, 2, 8, 4, 32, 32, 128, 32, 512, 512, 2048, 1024, 8192, 8192, 32768, 4096, 131072, 131072, 524288, 262144, 2097152, 2097152, 8388608, 2097152, 33554432, 33554432, 134217728, 67108864, 536870912, 536870912, 2147483648, 134217728, 8589934592, 8589934592, 34359738368, 17179869184, 137438953472

Offset: 1

Author

Reinhard Zumkeller, Sep 01 2002

Keywords

• nonn
• frac

Links

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

Crossrefs

Denominator is A000265(n).

Cf. A007814, A051179, A073113, A084623, A273893.

Programs

Magma

[Numerator(2^n/n): n in [1..50]]; // G. C. Greubel, Feb 28 2019

Maple

[seq(numer(2^n/n),n=1..50)];

Mathematica

f[n_]:=Numerator[2^n/n]; Array[f,50] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2011 *)

PARI

a(n) = numerator(2^n/n); \\ Michel Marcus, Mar 25 2018

PARI

a(n) = 2^(n - valuation(n, 2)) \\ Jianing Song, Oct 24 2018

Python

from fractions import Fraction
def A075101(n):
    return (Fraction(2**n)/n).numerator # Chai Wah Wu, Mar 25 2018

Sage

[numerator(2^n/n) for n in (1..50)] # G. C. Greubel, Feb 28 2019

Formula

a(n) = 2^(n - A007814(n)).

a(n) = 2*A084623(n). - Paul Curtz, Jan 28 2013

a(n) = 2^A093048(n). - Paul Curtz, Jun 10 2016

From Peter Bala, Feb 25 2019: (Start)

a(n) = 2^n/gcd(n,2^n).

O.g.f.: F(2*x) - (1/2)*F((2*x)^2) - (1/4)*F((2*x)^4) - (1/8)*F((2*x)^8) - ..., where F(x) = x/(1 - x). Cf. A000265.

O.g.f. for reciprocals: Sum_{n >= 1} x^n/a(n) = F((x/2)) + F((x/2)^2) + 2*F((x/2)^4) + 4*F((x/2)^8) + 8*F((x/2)^16) + 16*F((x/2)^32) + .... (End)

Sum_{n>=1} 1/a(n) = Sum_{n>=1} 2^(2^(n-1)+n-1)/(2^(2^n) - 1) = Sum_{n>=1} A073113(n-1)/A051179(n) = 1.48247501707... - Amiram Eldar, Aug 14 2022