A003071 Sorting numbers: maximal number of comparisons for sorting n elements by list merging. (Formerly M2443)

0, 1, 3, 5, 9, 11, 14, 17, 25, 27, 30, 33, 38, 41, 45, 49, 65, 67, 70, 73, 78, 81, 85, 89, 98, 101, 105, 109, 115, 119, 124, 129, 161, 163, 166, 169, 174, 177, 181, 185, 194, 197, 201, 205, 211, 215, 220, 225, 242, 245, 249, 253, 259, 263, 268, 273, 283, 287, 292, 297, 304

Offset

1,3

Comments

The following sequences all appear to have the same parity: A003071, A029886, A061297, A092524, A093431, A102393, A104258, A122248, A128975. - Jeremy Gardiner, Dec 28 2008

a(A092246(n)) = A230720(n); a(A230709(n)) = A230721(n+1). - Reinhard Zumkeller, Oct 28 2013

References

D. E. Knuth, Art of Computer Programming, Vol. 3, Sections 5.2.4 and 5.3.1.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Zak Seidov, Table of n, a(n) for n = 1..10000

J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.

J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.

An Vinh Nguyen Dinh, Nhien Pham Hoang Bao, Terrillon Jean-Christophe, Hiroyuki Iida, Reaper Tournament System, 2018.

An Vinh Nguyen Dinh, Nhien Pham Hoang Bao, Mohd Nor Akmal Khalid, Hiroyuki Iida, Simulating competitiveness and precision in a tournament structure: a reaper tournament system, Int'l J. of Information Technology (2020) Vol. 12, 1-18.

Tanya Khovanova, There are no coincidences, arXiv preprint 1410.2193 [math.CO], 2014.

Index entries for sequences related to sorting

Formula

Let n = 2^e_1 + 2^e_2 + ... + 2^e_t, e_1 > e_2 > ... > e_t >= 0, t >= 1. Then a(n) = 1 - 2^e_t + Sum_{k=1..t} (e_k + k - 1)*2^e_k [Knuth, Problem 14, Section 5.2.4].

a(n) = a(n-1) + A061338(n) = a(n-1) + A006519(n) + A000120(n) - 1 = n + A000337(A000523(n)) + a(n - 2^A000523(n)). a(2^k) = k*2^k + 1 = A002064(k). - Henry Bottomley, Apr 27 2001

G.f.: x/(1-x)^3 + 1/(1-x)^2*Sum(k>=1, (-1+(1-x)*2^(k-1))*x^2^k/(1-x^2^k)). - Ralf Stephan, Apr 17 2003

Mathematica

a[1] = 0; a[n_] := a[n] = a[n-1] + 2^IntegerExponent[n-1, 2] + DigitCount[n-1, 2, 1] - 1; Table[a[n], {n, 1, 61}] (* Jean-François Alcover, Feb 10 2012, after Henry Bottomley *)

Prog

(Haskell)
a003071 n = 1 - 2 ^ last es +
   sum (zipWith (*) (zipWith (+) es [0..]) (map (2 ^) es))
   where es = reverse $ a133457_row n
-- Reinhard Zumkeller, Oct 28 2013

Crossrefs

Cf. A001855, A133457.

Sequence in context: A139099 A152259 A219611 * A178442 A319986 A359408

Adjacent sequences: A003068 A003069 A003070 * A003072 A003073 A003074

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from David W. Wilson

Status

approved