A113474 a(n) = a(floor(n/2)) + floor(n/2) with a(1) = 1.

1, 2, 2, 4, 4, 5, 5, 8, 8, 9, 9, 11, 11, 12, 12, 16, 16, 17, 17, 19, 19, 20, 20, 23, 23, 24, 24, 26, 26, 27, 27, 32, 32, 33, 33, 35, 35, 36, 36, 39, 39, 40, 40, 42, 42, 43, 43, 47, 47, 48, 48, 50, 50, 51, 51, 54, 54, 55, 55, 57, 57, 58, 58, 64, 64, 65, 65, 67, 67, 68, 68, 71, 71

Offset: 1

Author

Zak Seidov, Jan 08 2006

Keywords

• nonn
• easy

Comments

a(2^n) = 2^n, in other cases a(n) < n. Except for the initial 1 all entries are repeated. Apparently no simple formula is known for a(n).

Taking every other term seems to give A101925. - Dominick Cancilla, Aug 03 2010

1/a(n) is the probability that a randomly chosen divisor of n! is odd. This is because the product n! contains the prime factor 2 a total of a(n) - 1 times (cf. A011371) and thus the prime factor 2 can occur in a divisor of n! a total of a(n) times, ranging between 0 and a(n) - 1 times. - Martin Renner, Dec 28 2022

Links

• G. C. Greubel, Table of n, a(n) for n = 1..2500
• Tanya Khovanova, There are no coincidences, arXiv:1410.2193 [math.CO], 2014.

Crossrefs

Cf. A005187, A011371, A101925.

Programs

Magma

A113474:= func< n | n+1-Multiplicity(Intseq(n, 2), 1) >;
[A113474(n): n in [1..90]]; // G. C. Greubel, Sep 28 2024

Mathematica

a[1]=1;a[n_]:=a[n]=a[Floor[n/2]]+Floor[n/2]; Table[a[n], {n, 100}]

PARI

for(n=1, 90, print1(1 - n + sum(k=0,n, n\2^k), ", ")) \\ G. C. Greubel, Mar 11 2017

PARI

a(n)=sum(k=1,logint(n,2), n>>k)+1 \\ Charles R Greathouse IV, Mar 12 2017

PARI

a(n)=n+1-hammingweight(n) \\ Charles R Greathouse IV, Dec 29 2022

SageMath

def A113474(n): return n+1 - sum((n+0).digits(2))
[A113474(n) for n in range(1,90)] # G. C. Greubel, Sep 28 2024

Formula

From Paul Barry, Aug 27 2006: (Start)

a(n) = ( Sum_{k=0..n} floor(n/2^k) ) - n + 1.

a(n) = 2 + Sum_{k=0..n} ( floor(n/2^k)-1 ).

a(n) = A005187(n) - n + 1. (End)

a(n) = n + O(log n). - Charles R Greathouse IV, Mar 12 2017

a(n) = A011371(n) + 1 for n > 0. - Martin Renner, Dec 28 2022