A011371 a(n) = n minus (number of 1's in binary expansion of n). Also highest power of 2 dividing n!.

0, 0, 1, 1, 3, 3, 4, 4, 7, 7, 8, 8, 10, 10, 11, 11, 15, 15, 16, 16, 18, 18, 19, 19, 22, 22, 23, 23, 25, 25, 26, 26, 31, 31, 32, 32, 34, 34, 35, 35, 38, 38, 39, 39, 41, 41, 42, 42, 46, 46, 47, 47, 49, 49, 50, 50, 53, 53, 54, 54, 56, 56, 57, 57, 63, 63, 64, 64, 66, 66, 67, 67, 70

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

Terms of A005187 repeated. - Lekraj Beedassy, Jul 06 2004

This sequence shows why in binary 0 and 1 are the only two numbers n such that n equals the sum of its digits raised to the consecutive powers (equivalent to the base-10 sequence A032799). 1 raised to any consecutive power is still 1 and thus any sum of digits raised to consecutive powers for any n > 1 falls short of equaling the value of n by the n-th term of this sequence. - Alonso del Arte, Jul 27 2004

Also the number of trailing zeros in the base-2 representation of n!. - Hieronymus Fischer, Jun 18 2007

Partial sums of A007814. - Philippe Deléham, Jun 21 2012

If n is in A089633 and n > 0, then a(n) = n - floor(log_2(n+1)). - Douglas Latimer, Jul 25 2012

For n > 1, denominators of integral numerator polynomials L(n,x) for the Legendre polynomials with o.g.f. 1/sqrt(1 - t*x + x^2). - Tom Copeland, Feb 04 2016

The definition of this sequence explains why, for n > 1, the highest power of 2 dividing n! added to the number of 1's in the binary expansion of n is equal to n. This result is due to the French mathematician Adrien Legendre (1752-1833) [see the Honsberger reference]. - Bernard Schott, Apr 07 2017

a(n) is the total number of 2's in the prime factorizations over the first n positive integers. The expected number of 2's in the factorization of an integer n is 1 (as n->infinity). Generally, the expected number of p's (for a prime p) is 1/(p-1). - Geoffrey Critzer, Jun 05 2017

Examples

			a(3) = 1 because 3 in binary is 11 (two 1's) and 3 - 2 = 1.
a(4) = 3 because 4 in binary is 100 (one 1 and two 0's) and 4 - 1 = 3.
a(5) = 3 because 5 in binary is 101 (a zero between two 1's) and 5 - 2 = 3.
a(100) = 97.
a(10^3) = 994.
a(10^4) = 9995.
a(10^5) = 99994.
a(10^6) = 999993.
a(10^7) = 9999992.
a(10^8) = 99999988.
a(10^9) = 999999987.
G.f. = x^2 + x^3 + 3*x^4 + 3*x^5 + 4*x^6 + 4*x^7 + 7*x^8 + 7*x^9 + 8*x^10 + ...
		

References

• K. Atanassov, On Some of Smarandache's Problems, section 7, on the 61st problem, page 42, American Research Press, 1999, 16-21.
• G. Bachman, Introduction to p-Adic Numbers and Valuation Theory, Academic Press, 1964; see Lemma 3.1.
• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 305.
• H. Davenport, The Higher Arithmetic, 7th ed. 1999, Cambridge University Press, p. 216, exercise 1.07.
• R. Honsberger, Mathematical Gems II, Dolciani Mathematical Expositions, 1976, pp. 1-6.

Links

• Hieronymus Fischer, Table of n, a(n) for n = 0..10000 (first 1000 terms from T. D. Noe)
• Laurent Alonso, Edward M. Reingold, and René Schott, Determining the majority, Inform. Process. Lett. 47 (1993), no. 5, 253-255.
• Laurent Alonso, Edward M. Reingold, and René Schott, The average-case complexity of determining the majority, SIAM J. Comput. 26 (1997), no. 1, 1-14.
• 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. - From _N. J. A. Sloane_, Dec 24 2012
• R. Hinze, Concrete stream calculus: An extended study, J. Funct. Progr. 20 (5-6) (2010) 463-535, doi, Section 4.4.
• Keith Johnson and Kira Scheibelhut, Rational polynomials that take integer values at the Fibonacci numbers, American Mathematical Monthly 123.4 (2016): 338-346. See omega_2.
• S-C Liu and J. C.-C. Yeh, Catalan numbers modulo 2^k, J. Int. Seq. 13 (2010), 10.5.4, eq. (5).
• K. Matthews, Computing the prime-power factorization of n!.
• A. Mir, F. Rossello and L. Rotger, A new balance index for phylogenetic trees, arXiv preprint arXiv:1202.1223 [q-bio.PE], 2012.
• A. M. Oller-Marcen and J. Maria Grau, On the Base-b Expansion of the Number of Trailing Zeros of b^k!, J. Int. Seq. 14 (2011) 11.6.8.
• Michael E. Saks and Michael Werman, On computing majority by comparisons, Combinatorica 11 (1991), no. 4, 383-387.
• Ralf Stephan, Some divide-and-conquer sequences ...
• Ralf Stephan, Table of generating functions.
• Zhujun Zhang, A Note on Counting Binomial Heaps, ResearchGate (2019).

Crossrefs

Cf. A000120, A005187, A054861, A032799, A067080, A098844, A132027.

a(n) = Sum_{k=1..n} A007814(k), n >= 1, a(0) = 0.

Programs

Haskell

a011371 n = n - a000120 n  -- Reinhard Zumkeller, Jan 24 2014

Magma

[Valuation(Factorial(n), 2): n in [0..80]]; // Bruno Berselli, Aug 05 2013

Maple

A011371(n) = RETURN(((2^(l))-1)+sum('(j*floor((n-(2^l)+2^j)/(2^(j+1))))','j'=1..l)); # after K. Atanassov. Here l is [ log2(n) ].
A011371 := n -> n - add(i,i=convert(n,base,2)): # Peter Luschny, May 02 2009
read("transforms") : A011371 := proc(n) n-wt(n) ; end proc: # R. J. Mathar, May 15 2013

Mathematica

-1 + Length[ Last[ Split[ IntegerDigits[ 2(n!), 2 ] ] ] ], FoldList[ Plus, 0, Fold[ Flatten[ {#1, #2, #1} ]&, 0, Range[ 6 ] ] ]
Table[IntegerExponent[n!, 2], {n, 0, 127}]
Table[n - DigitCount[n, 2, 1], {n, 0, 127}]
Table[t = 0; p = 2; While[s = Floor[n/p]; t = t + s; s > 0, p *= 2]; t, {n, 0, 100} ]

PARI

{a(n) = if( n<0, 0, valuation(n!, 2))}; /* Michael Somos, Oct 24 2002 */

PARI

{a(n) = if( n<0, 0, sum(k=1, n, n\2^k))}; /* Michael Somos, Oct 24 2002 */

PARI

{a(n) = if( n<0, 0, n - subst( Pol( binary( n ) ), x, 1))}; /* Michael Somos, Aug 28 2007 */

PARI

a(n)=sum(k=1,log(n+1)\log(2),n>>k) \\ Charles R Greathouse IV, Oct 03 2012

PARI

a(n)=my(s);while(n>>=1,s+=n);s \\ Charles R Greathouse IV, Aug 09 2013

PARI

a(n) = n - hammingweight(n); \\ Michel Marcus, Jun 05 2014

Python

[n - bin(n)[2:].count("1") for n in range(101)] # Indranil Ghosh, Apr 09 2017

Python

# 3.10+
def A011371(n): return n-n.bit_count() # Chai Wah Wu, Jul 09 2022

Formula

a(n) = a(floor(n/2)) + floor(n/2) = floor(n/2) + floor(n/4) + floor(n/8) + floor(n/16) + ... - Henry Bottomley, Apr 24 2001

G.f.: A(x) = (1/(1 - x))*Sum_{k>=1} x^(2^k)/(1 - x^(2^k)). - Ralf Stephan, Apr 11 2002

a(n) = n - A000120(n). - Lekraj Beedassy, Sep 01 2003

a(n) = A005187(n) - n, n >= 0.

a(n) = A007814(A000142(n)). - Reinhard Zumkeller, Apr 09 2004

From Hieronymus Fischer, Jun 25 and Aug 13 2007: (Start)

a(n) = Sum_{k=2..n} Sum_{j|k, j >= 2} (floor(log_2(j)) - floor(log_2(j - 1))).

The g.f. can be expressed in terms of a Lambert series, in that g(x) = L[b(k)](x)/(1 - x), where

L[b(k)](x) = Sum_{k>=0} b(k)*x^k/(1 - x^k) is a Lambert series with b(k) = 1, if k is a power of 2, otherwise b(k) = 0.

G.f.: g(x) = (1/(1-x))*Sum_{k>0} c(k)*x^k, where c(k) = Sum_{j>1, j|k} (floor(log_2(j)) - floor(log_2(j-1))).

Recurrence:

a(n) = floor(n/2) + a(floor(n/2));

a(2*n) = n + a(n);

a(n*2^m) = n*(2^m - 1) + a(n).

a(2^m) = 2^m - 1, m >= 0.

Asymptotic behavior:

a(n) = n + O(log(n)),

a(n+1) - a(n) = O(log(n)), which follows from the inequalities below.

a(n) <= n - 1; equality holds for powers of 2.

a(n) >= n - 1 - floor(log_2(n)); equality holds for n = 2^m - 1, m > 0.

lim inf (n - a(n)) = 1, for n->oo.

lim sup (n - log_2(n) - a(n)) = 0, for n->oo.

lim sup (a(n+1) - a(n) - log_2(n)) = 0, for n->oo. (End)

a(n) = Sum_{k >= 0} A030308(n, k)*A000225(k). - Philippe Deléham, Oct 16 2011

a(n) = Sum_{k=0..floor(log_2(n+1))} f^(k+1)(n), where f(n) = (n - (n mod 2))/2 and f^(k+1) is the (k+1)-th composition of f. - Joseph Wheat, Mar 01 2018

a(n) = Sum_{k=1..floor(n/2)} floor(log_2(n/k)). - Ammar Khatab, Feb 01 2025

Extensions

Examples added by Hieronymus Fischer, Jun 06 2012