A073113 -id:A073113 - 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

A352081 Numbers of the form k*p^k, where k>1 and p is a prime.

Original entry on oeis.org

8, 18, 24, 50, 64, 81, 98, 160, 242, 324, 338, 375, 384, 578, 722, 896, 1029, 1058, 1215, 1682, 1922, 2048, 2500, 2738, 3362, 3698, 3993, 4374, 4418, 4608, 5618, 6591, 6962, 7442, 8978, 9604, 10082, 10240, 10658, 12482, 13778, 14739, 15309, 15625, 15842, 18818

Offset: 1

Amiram Eldar, Apr 16 2022

nonn

Each term in this sequence has a single presentation in the form k*p^k.

			8 is a term since 8 = 2*2^2.
18 is a term since 18 = 2*3^2.
24 is a term since 24 = 3*2^3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Lindqvist and Jaak Peetre, On the remainder in a series of Mertens, Expositiones Mathematicae, Vol. 15 (1997), pp. 467-478. See eq. (3).
Eric Weisstein's World of Mathematics, Mertens Constant. See eq. (5).

Subsequences: A036289 \ {0, 2}, A036290 \ {0, 3}, A036291 \ {0, 5}, A036293 \ {0, 7}, A073113 \ {2}, A079704, A100042, A104126.

Cf. A001620, A077761, A143524.

addP[p_, n_] := Module[{k = 2, s = {}, m}, While[(m = k*p^k) <= n, k++; AppendTo[s, m]]; s]; seq[max_] := Module[{m = Floor[Sqrt[max/2]], s = {}, ps}, ps = Select[Range[m], PrimeQ]; Do[s = Join[s, addP[p, max]], {p, ps}]; Sort[s]]; seq[2*10^4]

Sum_{n>=1} 1/a(n) = -A143524 = gamma - B_1, where gamma is Euler's constant (A001620), and B_1 is Mertens's constant (A077761).

cp's OEIS Frontend

A075101 Numerator of 2^n/n.

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