%I A001511 M0127 N0051
%S A001511 1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,6,1,2,1,
%T A001511 3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,7,1,2,1,3,1,2,
%U A001511 1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,6,1,2,1,3,1,2,1,4,1
%N A001511 The ruler function: 2^a(n) divides 2n. Or, a(n) = 2-adic valuation of
2n.
%C A001511 a(n) is the number of digits that must be counted from right to left
to reach the first 1 in the binary representation of n. For example,
a(12)=3 digits must be counted from right to left to reach the first
1 in 1100, the binary representation of 12. - anon, May 17 2002
%C A001511 If you are counting in binary and the least significant bit is numbered
1, the next bit is 2, etc., a(n) is the bit that is incremented when
increasing from n-1 to n. - Jud McCranie, Apr 26, 2004
%C A001511 Number of steps to reach an integer starting with (n+1)/2 and using the
map x -> x*ceiling(x) (cf. A073524).
%C A001511 a(n) = number of disk to be moved at n-th step of optimal solution to
Tower of Hanoi problem (comment from Andreas M. Hinz (hinz(AT)appl-math.tu-muenchen.de)).
%C A001511 Shows which bit to flip when creating the binary reflected Gray code
(bits are numbered from the right, offset is 1). This is essentially
equivalent to Hinz's comment. - Adam Kertesz (adamkertesz(AT)worldnet.att.net),
Jul 28 2001
%C A001511 a(n) is the Hamming distance between n and n-1 (in binary). This is equivalent
to Kertesz's comments above. - Tak-Shing Chan (chan12(AT)alumni.usc.edu),
Feb 25 2003
%C A001511 Let S(0) = {1}, S(n) = {S(n-1), S(n-1)-{x}, x+1} where x = last term
of S(n-1); sequence gives S(infinity). - Benoit Cloitre (benoit7848c(AT)orange.fr),
Jun 14 2003
%C A001511 The sum of all terms up to and including the first occurrence of m is
2^m-1. - Donald Sampson (marsquo(AT)hotmail.com), Dec 01 2003
%C A001511 m appears every 2^m terms starting with the 2^(m-1)th term. - Donald
Sampson (marsquo(AT)hotmail.com), Dec 08 2003
%C A001511 Sequence read mod 4 gives A092412. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr),
Mar 28 2004
%C A001511 If q = 2n/2^A001511(n) and if b(m) is defined by b(0)=q-1 and b(m)=2*b(m-1)+1,
then 2n = b(A001511(n)) + 1. - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net),
Dec 18 2004
%C A001511 Repeating pattern ABACABADABACABAE ... - Jeremy Gardiner (jeremy.gardiner(AT)btinternet.com),
Jan 16 2005
%C A001511 Relation to C(n) = Collatz function iteration using only odd steps: a(n)
is the number of right bits set in binary representation of A004767(n)
(numbers of the form 4*m+3). So for m=A004767(n) it follows that
there are exactly a(n) recursive steps where m<C(m). - Lambert Klasen
(lambert.klasen(AT)gmx.de), Jan 23 2005
%C A001511 The ordinal transform of a sequence b_0, b_1, b_2, ... is the sequence
a_0, a_1, a_2, ... where a_n is the number of times b_n has occurred
in {b_0 ... b_n}.
%C A001511 Between every two instances of any positive integer m there are exactly
m distinct values (1 through m-1 and one value greater than m). -
Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Sep 18 2006
%C A001511 Number of divisors of n of the form 2^k. - Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it),
Jul 25 2007
%C A001511 Every subsequence through n = (2k - 1) is a palindrome. [From Gary W.
Adamson (qntmpkt(AT)yahoo.com), Sep 24 2008]
%C A001511 A001511(n) = A007814(n) + 1.
%C A001511 2*n = A001511 * A000265 [From Eric Desbiaux (moongerms(AT)wanadoo.fr),
May 14 2009]
%C A001511 Multiplicative with a(2^k) = k + 1, a(p^k) = 1 for any odd prime p. [From
Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Jun 09 2009]
%C A001511 1 interleaved with (2 interleaved with (3 interleaved with ( ... )))
[From Eric D. Burgess (ericdb(AT)gmail.com), Oct 17 2009]
%C A001511 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 26 2009:
(Start)
%C A001511 A054525 (Mobius transform) * A001511 = A036987, the Fredholm-Rueppel
%C A001511 sequence, (1, 1, 0, 1, 0, 0, 0, 1,...) = A047999^(-1) * A001511 = the
%C A001511 inverse of Sierpinski's gaskit * the ruler sequence. (End)
%C A001511 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 26 2009:
(Start)
%C A001511 Equals A051731 * A036987, (inverse Mobius transform of the Fredholm-Rueppel
%C A001511 sequence; = A047999 * A036987, (Sierpinski's gasket * A036987). (End)
%D A001511 J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical
Computer Sci., 98 (1992), 163-197.
%D A001511 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press,
NY, 2 vols., 2nd ed., 2001-2003; see Dim- and Dim+ on p. 98; Dividing
Rulers, on pp. 436-437; The Ruler Game, pp. 469-470; Ruler Fours,
Fives, ... Fifteens on p. 470.
%D A001511 Flajolet, P., Raoult, J.-C. and Vuillemin, J.; The number of registers
required for evaluating arithmetic expressions. Theoret. Comput.
Sci. 9 (1979), no. 1, 99-125.
%D A001511 F. Q. Gouvea, p-Adic Numbers, Springer-Verlag, 1993; see p. 23.
%D A001511 A. M. Hinz, The Tower of Hanoi, in Algebras and combinatorics (Hong Kong,
1997), 277-289, Springer, Singapore, 1999.
%D A001511 Problem 636, Math. Mag., 40 (1967), 164-165.
%D A001511 Andrew Schloss, "Towers of Hanoi" composition, in The Digital Domain.
Elektra/Asylum Records 9 60303-2, 1983. Works by Jaffe (Finale to
``Silicon Valley Breakdown''), McNabb (``Love in the Asylum''), Schloss
(``Towers of Hanoi''), Mattox (``Shaman''), Rush, Moorer (``Lions
are Growing'') and others.
%D A001511 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A001511 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A001511 Dinesh Thakur, J. Number Theory, 1991.
%H A001511 N. J. A. Sloane, <a href="b001511.txt">Table of n, a(n) for n = 1..10000</
a>
%H A001511 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Fxtbook</a>
%H A001511 J.-P. Allouche and J. Shallit, <a href="http://www.cs.uwaterloo.ca/~shallit/
Papers/as0.ps">The ring of k-regular sequences</a>, Theoretical Computer
Sci., 98 (1992), 163-197.
%H A001511 J. Britton, <a href="http://britton.disted.camosun.bc.ca/jbhanoi.htm">
Tower of Hanoi Solution</a>
%H A001511 J. C. Lagarias and N. J. A. Sloane, Approximate squaring (<a href="http:/
/www.research.att.com/~njas/doc/apsq.pdf">pdf</a>, <a href="http:/
/www.research.att.com/~njas/doc/apsq.ps">ps</a>), Experimental Math.,
13 (2004), 113-128.
%H A001511 Michael Naylor, <a href="http://www.ac.wwu.edu/~mnaylor/abacaba/abacaba.html">
Abacaba-Dabacaba</a>
%H A001511 R. Stephan, <a href="http://arXiv.org/abs/math.CO/0307027">Divide-and-conquer
generating functions. I. Elementary sequences</a>
%H A001511 R. Stephan, <a href="somedcgf.html">Some divide-and-conquer sequences
...</a>
%H A001511 R. Stephan, <a href="a079944.ps">Table of generating functions</a>
%H A001511 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
BinaryCarrySequence.html">Link to a section of The World of Mathematics.</
a>
%H A001511 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
RulerFunction.html">Link to a section of The World of Mathematics.</
a>
%H A001511 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
TowersofHanoi.html">Link to a section of The World of Mathematics.</
a>
%H A001511 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A001511 <a href="Sindx_Bi.html#binary">Index entries for sequences related to
binary expansion of n</a>
%F A001511 a(2n+1) = 1; a(2n) = 1 + a(n). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr),
Dec 08 2003
%F A001511 a(n) = 2-A000120(n)+A000120(n-1), n >= 1 - from Daniele Parisse (daniele.parisse(AT)m.dasa.de).
%F A001511 a(n) = 1 + lg(abs(A003188(n)-A003188(n-1))), where lg is the base-2 logarithm.
%F A001511 Multiplicative with a(p^e) = e+1 if p = 2; 1 if p > 2. - David W. Wilson
(davidwwilson(AT)comcast.net), Aug 01 2001.
%F A001511 For any real x > 1/2: lim N ->inf (1/N)*sum(n=1, N, x^(-a(n)))= 1/(2x-1);
also lim N ->inf (1/N)*Sum(n=1, N, 1/a(n))=ln(2). - Benoit Cloitre
(benoit7848c(AT)orange.fr), Nov 16 2001
%F A001511 s(n) = sum(k=1, n, a(k)) is asymptotic to 2*n since s(n)=2n-A000120(n).
- Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 31 2002
%F A001511 For any n>=0, for any m>=1, a(2^m*n+2^(m-1)) = m. - Benoit Cloitre, Nov
24 2002
%F A001511 a(n) = sum_{d divides n and d is odd} mu(d)*tau(n/d). - Vladeta Jovovic
(vladeta(AT)eunet.rs), Dec 04 2002
%F A001511 G.f.: A(x) = sum_{k=0..infinity} x^(2^k)/(1-x^(2^k)). - Ralf Stephan
(ralf(AT)ark.in-berlin.de), Dec 24 2002
%F A001511 a(1) = 1; for n > 1, a(n) = a(n-1)+(-1)^n*a(floor(n/2)). - Vladeta Jovovic
(vladeta(AT)eunet.rs), Apr 25 2003
%F A001511 Sum(k = 1 through n) a(k) = 2n - number of 1's in binary representation
of n. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 15 2003
%F A001511 A fixed point of the mapping 1->12; 2->13; 3->14; 4->15; 5->16; ....
- DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 13 2003
%F A001511 Product_{k>0}(1+x^k)^a(k) is g.f. for A000041(). - Vladeta Jovovic (vladeta(AT)eunet.rs),
Mar 26 2004
%F A001511 G.f. A(x) satisfies A(x) = A(x^2) + x/(1-x). - Frank Adams-Watters (FrankTAW(AT)Netscape.net),
Feb 09 2006
%F A001511 a(A118413(n,k))=A002260(n,k); = a(A118416(n,k))=A002024(n,k); a(A014480(n))=A003602(A014480(n)).
- Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 27 2006
%F A001511 Ordinal transform of A003602. - Frank Adams-Watters (FrankTAW(AT)Netscape.net),
Aug 28 2006
%F A001511 Could be extended to n <= 0 using a(-n)=a(n), a(0)=0, a(2n)=a(n)+1 unless
n=0. - Michael Somos Sep 30 2006
%F A001511 Sequence = A129360 * A000005 = M*V, where M = an infinite lower triangular
matrix and V = d(n) as a vector: [1, 2, 2, 3, 2, 4,...]. - Gary W.
Adamson (qntmpkt(AT)yahoo.com), Apr 15 2007
%F A001511 A001511 = row sums of triangle A130093. - Gary W. Adamson (qntmpkt(AT)yahoo.com),
May 13 2007
%F A001511 Dirichlet g.f.: zeta(s)*2^s/(2^s-1). - Ralf Stephan, Jun 17 2007
%F A001511 a(n)=sum_{d divides n} mu(2d)*tau(n/d). - Benoit Cloitre (benoit7848c(AT)orange.fr),
Jun 21 2007
%F A001511 G.f.: x/(1-x) = Sum_{n>=1} a(n)*x^n*( 1 - x^n ) . - Paul D. Hanna (pauldhanna(AT)juno.com),
Jun 22 2007
%e A001511 For example, 2^1|2, 2^2|4, 2^1|6, 2^3|8, 2^1|10, 2^2|12, ... giving the
initial terms 1, 2, 1, 3, 1, 2, ...
%e A001511 Contribution from Omar E. Pol (info(AT)polprimos.com), Jun 12 2009: (Start)
%e A001511 Triangle begins:
%e A001511 1;
%e A001511 2,1;
%e A001511 3,1,2,1;
%e A001511 4,1,2,1,3,1,2,1;
%e A001511 5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1;
%e A001511 6,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1;
%e A001511 7,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,6,1,2,
1,3,...
%e A001511 (End)
%p A001511 A001511 := n->2-wt(n)+wt(n-1); # where wt is defined in A000120
%p A001511 This is the binary logarithm of the denominator of (256^n-1)B_{8n}/n,
in Maple parlance a := n -> log[2](denom((256^n-1)*bernoulli(8*n)/
n)). [From Peter Luschny (peter(AT)luschny.de), May 31 2009]
%t A001511 Array[ If[ Mod[ #, 2] == 0, FactorInteger[ # ][[1, 2]], 0] &, 105] +
1 (* or *)
%t A001511 Nest[ Flatten[ # /. a_Integer -> {1, a + 1}] &, {1}, 7] (from Robert
G. Wilson v Mar 04 2005)
%o A001511 (PARI) a(n) = sum(k=0,floor(log(n)/log(2)),floor(n/2^k)-floor((n-1)/2^k))
(from R. Stephan)
%o A001511 (PARI) a(n)=if(n%2,1,factor(n)[1,2]+1) - Jon Perry (perry(AT)globalnet.co.uk),
Jun 06 2004
%o A001511 (PARI) {a(n)=if(n, valuation(n, 2)+1, 0)} /* Michael Somos Sep 30 2006
*/
%o A001511 (PARI) {a(n)=if(n==1,1,polcoeff(x-sum(k=1, n-1, a(k)*x^k*(1-x^k)*(1-x+x*O(x^n))),
n))} - Paul D. Hanna (pauldhanna(AT)juno.com), Jun 22 2007
%Y A001511 a(n) = A007814(n)+1, column 1 of table A050600. Cf. A018238. Sequence
read mod 2 gives A035263.
%Y A001511 From Marc LeBrun: A005187(n) = Sum a(k), k <= n.
%Y A001511 Cf. A003188, A065176, A050603, A007814, A007949, A005187, A085058, A089080.
%Y A001511 Sequence is bisection of A007814, A050603, A050604, A067029, A089309.
%Y A001511 A085058 is a bisection.
%Y A001511 Cf. A003278, A117303, A000005, A129360, A130093.
%Y A001511 A094267(2n)=A050603(2n)=A050603(2n+1)=a(n). - Michael Somos Sep 30 206.
%Y A001511 This is Guy Steele's sequence GS(4, 2) (see A135416).
%Y A001511 Cf. A000079. [From Omar E. Pol (info(AT)polprimos.com), Jun 12 2009]
%Y A001511 Cf. A054525, A047999 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct
26 2009]
%Y A001511 Cf. A051731 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 26 2009]
%Y A001511 Sequence in context: A157226 A156249 A164677 this_sequence A065704 A026100
A059127
%Y A001511 Adjacent sequences: A001508 A001509 A001510 this_sequence A001512 A001513
A001514
%K A001511 mult,nonn,nice,easy,core
%O A001511 1,2
%A A001511 N. J. A. Sloane (njas(AT)research.att.com).
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