A061168 Partial sums of floor(log_2(k)) (= A000523(k)).

0, 1, 2, 4, 6, 8, 10, 13, 16, 19, 22, 25, 28, 31, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98, 103, 108, 113, 118, 123, 128, 133, 138, 143, 148, 153, 158, 163, 168, 173, 178, 183, 188, 193, 198, 203, 208, 213, 218, 223, 228, 233, 238, 243, 248

Offset: 1

Author

Antti Karttunen, Apr 19 2001

Keywords

• nonn
• easy

Comments

Given a term b>0 of the sequence and its left hand neighbor c, the corresponding unique sequence index n with property a(n)=b can be determined by n(b)=e+(b-d*(e+1)+2*(e-1))/d, where d=b-c and e=2^d. - Hieronymus Fischer, Dec 05 2006

a(n) gives index of start of binary expansion of n in the binary Champernowne sequence A076478. - N. J. A. Sloane, Dec 14 2017

a(n) is the number of pairs in ancestor relationship (= transitive closure of the parent relationship) in all (binary) heaps on n elements. - Alois P. Heinz, Feb 13 2019

References

• D. E. Knuth, Fundamental Algorithms, Addison-Wesley, 1973, Section 1.2.4, ex. 42(b).

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)
• Jean-Paul Allouche and Jeffrey Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197, ex. 27.
• Sung-Hyuk Cha, On Integer Sequences Derived from Balanced k-ary Trees, Applied Mathematics in Electrical and Computer Engineering, 2012.
• Sung-Hyuk Cha, On Complete and Size Balanced k-ary Tree Integer Sequences, International Journal of Applied Mathematics and Informatics, Issue 2, Volume 6, 2012, pp. 67-75.
• Alan M. Cleary, Joseph Winjum, Jordan Dood, Hiroki Shibata, and Shunsuke Inenaga, Bit Packed Encodings for Grammar-Compressed Strings Supporting Fast Random Access, Leibniz Int'l Proc. Informatics (LIPIcs) 23rd Int'l Symp. on Experim. Algor. (SEA 2025) Art. No. 12, 12:1-12:17. See p. 12:8.
• Martin Griffiths, More sums involving the floor function, Math. Gaz., 86 (2002), 285-287.
• Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.
• Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
• Tamás Lengyel, On the 2-Adic Valuation of Differences of Harmonic Numbers, Integers (2024) Vol. 24, A27. See p. 8.
• Eric Weisstein's World of Mathematics, Heap
• Wikipedia, Binary heap

Crossrefs

Cf. A000523, A001855, A123753, A076478.

Programs

Haskell

import Data.List (transpose)
a061168 n = a061168_list !! n
a061168_list = zipWith (+) [0..] (zipWith (+) hs $ tail hs) where
   hs = concat $ transpose [a001855_list, a001855_list]
-- Reinhard Zumkeller, Jun 03 2013

Maple

seq(add(floor(log[2](k)),k=1..j),j=1..100);
# second Maple program:
a:= proc(n) option remember; `if`(n<1, 0, ilog2(n)+a(n-1)) end:
seq(a(n), n=1..80);   # Alois P. Heinz, Feb 12 2019

Mathematica

Accumulate[Floor[Log[2,Range[110]]]] (* Harvey P. Dale, Jul 16 2012 *)
a[n_] := (n+1) IntegerLength[n+1, 2] - 2^IntegerLength[n+1, 2] - n + 1;
Table[a[n], {n, 1, 61}] (* Peter Luschny, Dec 02 2017 *)

PARI

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

PARI

a(n)=local(k); if(n<1,0,k=length(binary(n))-1; (n+1)*k-2*(2^k-1))

PARI

{ for (n=1, 1000, k=length(binary(n))-1; write("b061168.txt", n, " ", (n + 1)*k - 2*(2^k - 1)) ) } \\ Harry J. Smith, Jul 18 2009

Python

def A061168(n):
    s, i, z = -n , n, 1
    while 0 <= i: s += i; i -= z; z += z
    return s
print([A061168(n) for n in range(1, 62)]) # Peter Luschny, Nov 30 2017

Python

def A061168(n): return (n+1)*((m:=n.bit_length())-1)-(1<Chai Wah Wu, Mar 29 2023

Formula

a(n) = A001855(n+1) - n.

a(n) = Sum_{k=1..n} floor(log_2(k)) = (n+1)*floor(log_2(n)) - 2*(2^floor(log_2(n)) - 1). - Diego Torres (torresvillarroel(AT)hotmail.com), Oct 29 2002

G.f.: 1/(1-x)^2 * Sum(k>=1, x^2^k). - Ralf Stephan, Apr 13 2002

a(n) = A123753(n) - 2*n - 1. - Peter Luschny, Nov 30 2017