A093048 -id:A093048 - cp's OEIS frontend

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

Reinhard Zumkeller, Sep 01 2002

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

Denominator is A000265(n).

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

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

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

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

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

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

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

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

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

A093049 n-1 minus exponent of 2 in n, a(0) = 0.

Original entry on oeis.org

0, 0, 0, 2, 1, 4, 4, 6, 4, 8, 8, 10, 9, 12, 12, 14, 11, 16, 16, 18, 17, 20, 20, 22, 20, 24, 24, 26, 25, 28, 28, 30, 26, 32, 32, 34, 33, 36, 36, 38, 36, 40, 40, 42, 41, 44, 44, 46, 43, 48, 48, 50, 49, 52, 52, 54, 52, 56, 56, 58, 57, 60, 60, 62, 57, 64, 64, 66, 65, 68

Offset: 0

Ralf Stephan, Mar 16 2004

nonn

			G.f. = 2*x^3 + x^4 + 4*x^5 + 4*x^6 + 6*x^7 + 4*x^8 + 8*x^9 + 8*x^10 + ... - _Michael Somos_, Jan 25 2020

a(n) = n - A007814(n) - 1 = A093048(n) - 1, n>0.

a(n) is the exponent of 2 in A001761(n+1), A002105(n), A002682(n-1), A006963(n), A036770(n-1), A059837(n), A084623(n), |A003707(n)|, |A011859(n)|.

a[ n_] := If[ n == 0, 0, n - 1 - IntegerExponent[n, 2]]; (* Michael Somos, Jan 25 2020 *)

a(n)=if(n<1,0,if(n%2==0,a(n/2)+n/2-1,n-1))

{a(n) = if( n, n - 1 - valuation(n, 2))}; /* Michael Somos, Jan 25 2020 */

def A093049(n): return n-1-(~n& n-1).bit_length() if n else 0 # Chai Wah Wu, Jul 07 2022

Recurrence: a(2n) = a(n) + n - 1, a(2n+1) = 2n.

G.f.: sum(k>=0, t^3(t+2)/(1-t^2)^2, t=x^2^k).

A273893 Denominator of n/3^n.

Original entry on oeis.org

1, 3, 9, 9, 81, 243, 243, 2187, 6561, 2187, 59049, 177147, 177147, 1594323, 4782969, 4782969, 43046721, 129140163, 43046721, 1162261467, 3486784401, 3486784401, 31381059609, 94143178827, 94143178827, 847288609443, 2541865828329, 282429536481, 22876792454961

Offset: 0

Paul Curtz, Jun 02 2016

The reduced values are Ms(n) = 0, 1/3, 2/9, 1/9, 4/81, 5/243, 2/243, 7/2187, 8/6561, 1/2187, ... .
Numerators: 0, 1, 2, 1, 4, ... = A038502(n).
Ms(-n) = 0, -3, -18, ... = - A036290(n).
Difference table of Ms(n):
0,          1/3,    2/9,    1/9,   4/81, 5/243, 2/243, ...
1/3,       -1/9,   -1/9,  -5/81, -7/243, -1/81, ...
-4/9,         0,   4/81,  8/243,  4/243, ...
4/9,       4/81, -4/243, -4/243, ...
-32/81, -16/243,      0, ...
80/243,  16/243, ...
-64/243, ...
etc.
The difference table of O(n) = n/2^n (Oresme numbers) has its 0's on the main diagonal. Here the 0's appear every two rows. For n/4^n,they appear every three rows. (The denominators of O(n) are 2^A093048(n)).
All terms are powers of 3 (A000244).

Cf. A007949, A000244, A013732, A013733, A036290, A038502, A075101.

Table[Denominator[n/3^n], {n, 0, 28}] (* Michael De Vlieger, Jun 03 2016 *)

a(n) = denominator(n/3^n) \\ Felix Fröhlich, Jun 07 2016

[1] + [3^(n-n.valuation(3)) for n in [1..30]] # Tom Edgar, Jun 02 2016

For n>0, a(n) = 3^(n - valuation(n,3)) = 3^(n - A007949(n)). - Tom Edgar, Jun 02 2016
a(3n+1) = 3^(3n+1), a(3n+2) = 3^(3n+2).
a(3n+6) = 27*(3n+3).
From Peter Bala, Feb 25 2019: (Start)
a(n) = 3^n/gcd(n,3^n).
O.g.f.: 1 + F(3*x) - (2/3)*F((3*x)^3) - (2/9)*F((3*x)^9) - (2/27)*F((3*x)^27) - ..., where F(x) = x/(1 - x).
O.g.f. for reciprocals: Sum_{n >= 0} x^n/a(n) = 1 +  F((x/3)) + 2*( F((x/3)^3) + 3*F((x/3)^9) + 9*F((x/3)^27) + ... ). Cf. A038502. (End)

cp's OEIS Frontend

A075101 Numerator of 2^n/n.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Python

Sage

Formula

A093049 n-1 minus exponent of 2 in n, a(0) = 0.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Formula

A273893 Denominator of n/3^n.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Sage

Formula