A073524 -id:A073524 - cp's OEIS frontend

A001511 The ruler function: exponent of the highest power of 2 dividing 2n. Equivalently, the 2-adic valuation of 2n.

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, 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, 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

Offset: 1

N. J. A. Sloane

Number of 2's dividing 2*n.
a(n) is equivalently the exponent of the smallest power of 2 which does not divide n. - David James Sycamore, Oct 02 2023
a(n) - 1 is the number of trailing zeros in the binary expansion of n.
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
Number of steps to reach an integer starting with (n+1)/2 and using the map x -> x*ceiling(x) (cf. A073524).
a(n) is the number of the disk to be moved at the n-th step of the optimal solution to Towers of Hanoi problem (comment from Andreas M. Hinz).
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, Jul 28 2001
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
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, Jun 14 2003
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
m appears every 2^m terms starting with the 2^(m-1)th term. - Donald Sampson (marsquo(AT)hotmail.com), Dec 08 2003
Sequence read mod 4 gives A092412. - Philippe Deléham, Mar 28 2004
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, Dec 18 2004
Repeating pattern ABACABADABACABAE ... - Jeremy Gardiner, Jan 16 2005
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 mBetween 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, Sep 18 2006
Number of divisors of n of the form 2^k. - Giovanni Teofilatto, Jul 25 2007
Every prefix up to (but not including) the first occurrence of some k >= 2 is a palindrome. - Gary W. Adamson, Sep 24 2008
1 interleaved with (2 interleaved with (3 interleaved with ( ... ))). - Eric D. Burgess (ericdb(AT)gmail.com), Oct 17 2009
A054525 (Möbius transform) * A001511 = A036987 = A047999^(-1) * A001511. - Gary W. Adamson, Oct 26 2009
Equals A051731 * A036987, (inverse Möbius transform of the Fredholm-Rueppel sequence) = A047999 * A036987. - Gary W. Adamson, Oct 26 2009
Cf. A173238, showing links between generalized ruler functions and A000041. - Gary W. Adamson, Feb 14 2010
Given A000041, P(x) = A(x)/A(x^2) with P(x) = (1 + x + 2x^2 + 3x^3 + 5x^4 + 7x^5 + ...), A(x) = (1 + x + 3x^2 + 4x^3 + 10x^4 + 13x^5 + ...), A(x^2) = (1 + x^2 + 3x^4 + 4x^6 + 10x^8 + ...), where A092119 = (1, 1, 3, 4, 10, ...) = Euler transform of the ruler sequence, A001511. - Gary W. Adamson, Feb 11 2010
Subtracting 1 from every term and deleting any 0's yields the same sequence, A001511. - Ben Branman, Dec 28 2011
In the listing of the compositions of n as lists in lexicographic order, a(k) is the last part of composition(k) for all k <= 2^(n-1) and all n, see example. - Joerg Arndt, Nov 12 2012
According to Hinz, et al. (see links), this sequence was studied by Louis Gros in his 1872 pamphlet "Théorie du Baguenodier" and has therefore been called the Gros sequence.
First n terms comprise least squarefree word of length n using positive integers, where "squarefree" means that the word contains no consecutive identical subwords; e.g., 1 contains no square; 11 contains a square but 12 does not; 121 contains no square; both 1211 and 1212 have squares but 1213 does not; etc. - Clark Kimberling, Sep 05 2013
Length of 0-run starting from 2 (10, 100, 110, 1000, 1010, ...), or length of 1-run starting from 1 (1, 11, 101, 111, 1001, 1011, ...) of every second number, from right to left in binary representation. - Armands Strazds, Apr 13 2017
a(n) is also the frequency of the largest part in the integer partition having viabin number n. The viabin number of an integer partition is defined in the following way. Consider the southeast border of the Ferrers board of the integer partition and consider the binary number obtained by replacing each east step with 1 and each north step, except the last one, with 0. The corresponding decimal form is, by definition, the viabin number of the given integer partition. "Viabin" is coined from "via binary". For example, consider the integer partition [2,2,2,1]. The southeast border of its Ferrers board yields 10100, leading to the viabin number 20. - Emeric Deutsch, Jul 24 2017
As A000005(n) equals the number of even divisors of 2n and A001227(n) = A001227(2n), the formula A001511(n) = A000005(n)/A001227(n) might be read as "The number of even divisors of 2n is always divisible by the number of odd divisors of 2n" (where number of divisors means sum of zeroth powers of divisors). Conjecture: For any nonnegative integer k, the sum of the k-th powers of even divisors of n is always divisible by the sum of the k-th powers of odd divisors of n. - Ivan N. Ianakiev, Jul 06 2019
From Benoit Cloitre, Jul 14 2022: (Start)
To construct the sequence, start from 1's separated by a place 1,,1,,1,,1,,1,,1,,1,,1,,1,,1,,1,,1,,1,,1,...
Then put the 2's in every other remaining place
  1,2,1,,1,2,1,,1,2,1,,1,2,1,,1,2,1,,1,2,1,,1,2,1,...
Then the 3's in every other remaining place
  1,2,1,3,1,2,1,,1,2,1,3,1,2,1,,1,2,1,3,1,2,1,,1,2,1,...
Then the 4's in every other remaining place
  1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,,1,2,1,3,1,2,1,4,1,2,1,...
By iterating this process, we get the ruler function 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,... (End)
a(n) is the least positive integer k for which there does not exist i+j=n and a(i)=a(j)=k (cf. A322523). - Rémy Sigrist and Jianing Song, Aug 23 2022
a(n) is the smallest positive integer that does not occur in the coincidences of the sequence so far a(1..n-1) and its reverse. - Neal Gersh Tolunsky, Jan 18 2023
The geometric mean of this sequence approaches the Somos constant (A112302). - Jwalin Bhatt, Jan 31 2025

			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, ...
From _Omar E. Pol_, Jun 12 2009: (Start)
Triangle begins:
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;
6,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1;
7,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,6,1,2,1,3,...
(End)
S(0) = {} S(1) = 1 S(2) = 1, 2, 1 S(3) = 1, 2, 1, 3, 1, 2, 1 S(4) = 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1. - Yann David (yann_david(AT)hotmail.com), Mar 21 2010
From _Joerg Arndt_, Nov 12 2012: (Start)
The 16 compositions of 5 as lists in lexicographic order:
[ n]  a(n)  composition
[ 1]  [ 1]  [ 1 1 1 1 1 ]
[ 2]  [ 2]  [ 1 1 1 2 ]
[ 3]  [ 1]  [ 1 1 2 1 ]
[ 4]  [ 3]  [ 1 1 3 ]
[ 5]  [ 1]  [ 1 2 1 1 ]
[ 6]  [ 2]  [ 1 2 2 ]
[ 7]  [ 1]  [ 1 3 1 ]
[ 8]  [ 4]  [ 1 4 ]
[ 9]  [ 1]  [ 2 1 1 1 ]
[10]  [ 2]  [ 2 1 2 ]
[11]  [ 1]  [ 2 2 1 ]
[12]  [ 3]  [ 2 3 ]
[13]  [ 1]  [ 3 1 1 ]
[14]  [ 2]  [ 3 2 ]
[15]  [ 1]  [ 4 1 ]
[16]  [ 5]  [ 5 ]
a(n) is the last part in each list.
(End)
From _Omar E. Pol_, Aug 20 2013: (Start)
Also written as a triangle in which the right border gives A000027 and row lengths give A011782 and row sums give A000079 the sequence begins:
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;
1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,7;
(End)
G.f. = x + 2*x^2 + x^3 + 3*x^4 + x^5 + 2*x^6 + x^7 + 4*x^8 + x^9 + 2*x^10 + ...

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003.
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.
L. Gros, Théorie du Baguenodier, Aimé Vingtrinier, Lyon, 1872.
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
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Antonin Callard, Léo Paviet Salomon, and Pascal Vanier, Computability of extender sets in multidimensional subshifts, arXiv:2401.07549 [cs.DM], 2024.
Imanuel Chen and Michael Z. Spivey, Integral Generalized Binomial Coefficients of Multiplicative Functions, Preprint 2015; Summer Research Paper 238, Univ. Puget Sound.
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Eric Weisstein's World of Mathematics, Binary Carry Sequence, Ruler Function, and Towers of Hanoi
Index entries for "core" sequences
Index entries for sequences related to binary expansion of n
Index entries for sequences that are fixed points of mappings
Index entries for sequences related to Towers of Hanoi

Column 1 of table A050600.
Sequence read mod 2 gives A035263.
Sequence is bisection of A007814, A050603, A050604, A067029, A089309.
This is Guy Steele's sequence GS(4, 2) (see A135416).
Cf. A005187 (partial sums), A085058 (bisection), A112302 (geometric mean), A171977 (2^a(n)).
Cf. A000005, A000041, A000079, A003188, A003278, A003602, A007949, A018238, A047999, A051731, A054525, A054852, A065176, A082850, A089080, A092119, A117303, A129360, A130093, A173238, A181988, A220466.
Cf. A287896, A002487, A209229 (Mobius trans.), A092673 (Dirichlet inv.).
Cf. generalized ruler functions for k=3,4,5: A051064, A115362, A055457.

a001511 n = length $ takeWhile ((== 0) . (mod n)) a000079_list
-- Reinhard Zumkeller, Sep 27 2011

a001511 n | odd n = 1 | otherwise = 1 + a001511 (n `div` 2)
-- Walt Rorie-Baety, Mar 22 2013

nmax=5;r=1;for n=2:nmax;r=[r n r];end % Adriano Caroli, Feb 26 2016

[Valuation(2*n,2): n in [1..105]]; // Bruno Berselli, Nov 23 2015

A001511 := n->2-wt(n)+wt(n-1); # where wt is defined in A000120
# 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)). - Peter Luschny, May 31 2009
A001511 := n -> padic[ordp](2*n,2): seq(A001511(n), n=1..105);  # Peter Luschny, Nov 26 2010
a:= n-> ilog2((Bits[Xor](2*n, 2*n-1)+1)/2): seq(a(n), n=1..50);  # Gary Detlefs, Dec 13 2018

Array[ If[ Mod[ #, 2] == 0, FactorInteger[ # ][[1, 2]], 0] &, 105] + 1 (* or *)
Nest[ Flatten[ # /. a_Integer -> {1, a + 1}] &, {1}, 7] (* Robert G. Wilson v, Mar 04 2005 *)
IntegerExponent[2*n, 2] (* Alexander R. Povolotsky, Aug 19 2011 *)
myHammingDistance[n_, m_] := Module[{g = Max[m, n], h = Min[m, n]}, b1 = IntegerDigits[g, 2]; b2 = IntegerDigits[h, 2, Length[b1]]; HammingDistance[b1, b2]] (* Vladimir Shevelev A206853 *) Table[ myHammingDistance[n, n - 1], {n, 111}] (* Robert G. Wilson v, Apr 05 2012 *)
Table[Position[Reverse[IntegerDigits[n,2]],1,1,1],{n,110}]//Flatten (* Harvey P. Dale, Aug 18 2017 *)

a(n) = sum(k=0,floor(log(n)/log(2)),floor(n/2^k)-floor((n-1)/2^k)) /* Ralf Stephan */

a(n)=if(n%2,1,factor(n)[1,2]+1) /* Jon Perry, Jun 06 2004 */

{a(n) = if( n, valuation(n, 2) + 1, 0)}; /* Michael Somos, Sep 30 2006 */

{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, Jun 22 2007 */

def a(n): return bin(n)[2:][::-1].index("1") + 1 # Indranil Ghosh, May 11 2017

A001511 = lambda n: (n&-n).bit_length() # M. F. Hasler, Apr 09 2020

def A001511(n): return (~n & n-1).bit_length()+1 # Chai Wah Wu, Jul 01 2022

[valuation(2*n,2) for n in (1..105)]  # Bruno Berselli, Nov 23 2015

(define (A001511 n) (let loop ((n n) (e 1)) (if (odd? n) e (loop (/ n 2) (+ 1 e))))) ;; Antti Karttunen, Oct 06 2017

a(n) = A007814(n) + 1 = A007814(2*n).
a(2*n+1) = 1; a(2*n) = 1 + a(n). - Philippe Deléham, Dec 08 2003
a(n) = 2 - A000120(n) + A000120(n-1), n >= 1. - Daniele Parisse
a(n) = 1 + log_2(abs(A003188(n) - A003188(n-1))).
Multiplicative with a(p^e) = e+1 if p = 2; 1 if p > 2. - David W. Wilson, Aug 01 2001
For any real x > 1/2: lim_{N->infinity} (1/N)*Sum_{n=1..N} x^(-a(n)) = 1/(2*x-1); also lim_{N->infinity} (1/N)*Sum_{n=1..N} 1/a(n) = log(2). - Benoit Cloitre, Nov 16 2001
s(n) = Sum_{k=1..n} a(k) is asymptotic to 2*n since s(n) = 2*n - A000120(n). - Benoit Cloitre, Aug 31 2002
For any n >= 0, for any m >= 1, a(2^m*n + 2^(m-1)) = m. - Benoit Cloitre, Nov 24 2002
a(n) = Sum_{d divides n and d is odd} mu(d)*tau(n/d). - Vladeta Jovovic, Dec 04 2002
G.f.: A(x) = Sum_{k>=0} x^(2^k)/(1-x^(2^k)). - Ralf Stephan, Dec 24 2002
a(1) = 1; for n > 1, a(n) = a(n-1) + (-1)^n*a(floor(n/2)). - Vladeta Jovovic, Apr 25 2003
A fixed point of the mapping 1->12; 2->13; 3->14; 4->15; 5->16; ... . - Philippe Deléham, Dec 13 2003
Product_{k>0} (1+x^k)^a(k) is g.f. for A000041(). - Vladeta Jovovic, Mar 26 2004
G.f. A(x) satisfies A(x) = A(x^2) + x/(1-x). - Franklin T. Adams-Watters, Feb 09 2006
a(A118413(n,k)) = A002260(n,k); = a(A118416(n,k)) = A002024(n,k); a(A014480(n)) = A003602(A014480(n)). - Reinhard Zumkeller, Apr 27 2006
Ordinal transform of A003602. - Franklin T. Adams-Watters, Aug 28 2006 (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}.)
Could be extended to n <= 0 using a(-n) = a(n), a(0) = 0, a(2*n) = a(n)+1 unless n=0. - Michael Somos, Sep 30 2006
A094267(2*n) = A050603(2*n) = A050603(2*n + 1) = a(n). - Michael Somos, Sep 30 2006
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, Apr 15 2007
Row sums of triangle A130093. - Gary W. Adamson, May 13 2007
Dirichlet g.f.: zeta(s)*2^s/(2^s-1). - Ralf Stephan, Jun 17 2007
a(n) = -Sum_{d divides n} mu(2*d)*tau(n/d). - Benoit Cloitre, Jun 21 2007
G.f.: x/(1-x) = Sum_{n>=1} a(n)*x^n*( 1 - x^n ). - Paul D. Hanna, Jun 22 2007
2*n = 2^a(n)* A000265(n). - Eric Desbiaux, May 14 2009 [corrected by Alejandro Erickson, Apr 17 2012]
Multiplicative with a(2^k) = k + 1, a(p^k) = 1 for any odd prime p. - Franklin T. Adams-Watters, Jun 09 2009
With S(n): 2^n - 1 first elements of the sequence then S(0) = {} (empty list) and if n > 0, S(n) = S(n-1), n, S(n-1). - Yann David (yann_david(AT)hotmail.com), Mar 21 2010
a(n) = log_2(A046161(n)/A046161(n-1)). - Johannes W. Meijer, Nov 04 2012
a((2*n-1)*2^p) = p+1, p >= 0 and n >= 1. - Johannes W. Meijer, Feb 05 2013
a(n+1) = 1 + Sum_{j=0..ceiling(log_2(n+1))} (j * (1 - abs(sign((n mod 2^(j + 1)) - 2^j + 1)))). - Enrico Borba, Oct 01 2015
Conjecture: a(n) = A181988(n)/A003602(n). - L. Edson Jeffery, Nov 21 2015
a(n) = log_2(A006519(n)) + 1. - Doug Bell, Jun 02 2017
Inverse Moebius transform of A209229. - Andrew Howroyd, Aug 04 2018
a(n) = 1 + (A183063(n)/A001227(n)). - Omar E. Pol, Nov 06 2018 (after Franklin T. Adams-Watters)
a(n) = log_2((Xor(2*n,2*n-1)+1)/2). - Gary Detlefs, Dec 13 2018
(2^(a(n)-1)-1)*(n mod 4) = 2*floor(((n+1) mod 4)/3). - Gary Detlefs, Dec 14 2018
a(n) = A000005(n)/A001227(n). - Ivan N. Ianakiev, Jul 05 2019
a(n) = Sum_{j=1..r} (j/2^j)*(Product_{k=1..j} (1 - (-1)^floor( (n+2^(j-1))/2^(k-1) ))), for n < a predefined 2^r. - Adriano Caroli, Sep 30 2019

Name edited following suggestion by David James Sycamore, Oct 05 2023

A068119 Number of steps to reach an integer starting with n + 1/4 and iterating the map x -> x*ceiling(x).

Original entry on oeis.org

3, 3, 1, 3, 2, 2, 1, 7, 4, 4, 1, 2, 2, 4, 1, 6, 3, 5, 1, 5, 2, 2, 1, 4, 6, 3, 1, 2, 2, 3, 1, 7, 3, 4, 1, 3, 2, 2, 1, 7, 4, 7, 1, 2, 2, 5, 1, 3, 3, 10, 1, 4, 2, 2, 1, 3, 5, 11, 1, 2, 2, 3, 1, 5, 3, 3, 1, 3, 2, 2, 1, 4, 4, 6, 1, 2, 2, 4, 1, 4, 3, 6, 1, 6, 2, 2, 1, 6, 7, 3, 1, 2, 2, 3, 1, 4, 3, 5, 1, 3, 2, 2, 1, 4

Offset: 1

Benoit Cloitre, Aug 30 2002

nonn

If the initial value is n + 1/2 we get A001511.
S(n)=sum(k=1, n, a(k)) seems to be asymptotic to 3*n. S(n)=3n for in A074069.
The sign of 3n-S(n) seems to change often: 3n-S(n) gives A074077. Is 3n-S(n) bounded? - Benoit Cloitre, Sep 05 2002

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.

Cf. A073524, A074069, A074077

import Data.Ratio ((%), denominator)
a068119 n = fst $ until ((== 1) . denominator . snd)
                        (\(i, x) -> (i + 1, f x)) (0, fromInteger n + 1%4)
   where f x = x * fromIntegral (ceiling x)
-- Reinhard Zumkeller, May 26 2013

ce[n_] := Length[NestWhileList[#*Ceiling[#] &, n + 1/4, ! IntegerQ[#] &]] - 1; ce /@ Range[104] (* Jayanta Basu, Jul 29 2013 *)

a(n)=if(n<0,0,s=n+1/4; c=0; while(frac(s)>0,s=s*ceil(s); c++); c)

a(n) = 1 if n == 3 (mod 4); a(n) = 2 if n == 5, 6, 12, 13 (mod 16); a(n) = 3 if n == 1, 2, 4, 17, 26, 30, 33, 36, 48, 49, 56, 62 (mod 64);...

Corrected by Diego Torres (torresvillarroel(AT)hotmail.com), Aug 31 2002

A072340 Number of steps to reach an integer starting with n/3 and iterating the map x -> x*ceiling(x), or -1 if no integer is ever reached.

Original entry on oeis.org

0, 2, 6, 0, 1, 1, 0, 5, 2, 0, 3, 2, 0, 1, 1, 0, 2, 4, 0, 3, 4, 0, 1, 1, 0, 22, 7, 0, 2, 5, 0, 1, 1, 0, 7, 2, 0, 4, 2, 0, 1, 1, 0, 2, 5, 0, 13, 9, 0, 1, 1, 0, 3, 3, 0, 2, 3, 0, 1, 1, 0, 3, 2, 0, 5, 2, 0, 1, 1, 0, 2, 3, 0, 8, 3, 0, 1, 1, 0, 5, 4, 0, 2, 4, 0, 1, 1, 0, 14, 2, 0, 3, 2, 0, 1, 1, 0, 2, 9, 0, 3, 9, 0, 1, 1

Offset: 3

N. J. A. Sloane and J. C. Lagarias, Sep 03 2002

nonn

We conjecture that an integer is always reached.

The occurrence of the first 1, 2, 3, 4 etc. is at the indices 7, 4, 13, 20, 10, 5, 29, 76, 50, 452, 244, 830, 49, 91, 319, 2639, 5753, 2215, 6151, 7148, 280, 28, 1783 - R. J. Mathar, Nov 25 2006

N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.

R. J. Mathar, Table of n, a(n) for n = 3..7147
J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.
N. J. A. Sloane, Seven Staggering Sequences.

Cf. A085276, A085285, A085286, A073524.

g := proc(x) local M,t1,t2,t3; M := 3^100; t1 := ceil(x) mod M; t2 := x*t1; t3 := numer(t2) mod M; t3/denom(t2); end;
f := proc(n) local t1,c; global g; if type(n, 'integer') then RETURN(0); fi; t1 := g(n); c := 1; while not type(t1, 'integer') do c := c+1; t1 := g(t1); od; RETURN(c); end;
[seq(f(n/3),n=3..120)]; # this gives the correct answer as long as the answer is < 99.

a[n_] := Module[{x = n/3, s = 0}, While[!IntegerQ[x], x *= Ceiling[x]; s++]; s]; Table[a[n], {n, 3, 107}] (* Jean-François Alcover, Jan 27 2019, from PARI *)

A072340(n)={ local(x,s) ; x=n/3 ; s=0 ; while( type(x)!="t_INT", x *= ceil(x) ; s++ ; ) ; return(s) ; } { for(n=3,10000, print(n," ",A072340(n)) ; ) ; } \\ R. J. Mathar, Nov 25 2006

A073341 Number of steps to reach an integer starting with (2n+1)/4 and iterating the map x -> x*ceiling(x).

Original entry on oeis.org

3, 2, 3, 8, 1, 1, 3, 2, 2, 3, 2, 2, 1, 1, 7, 4, 4, 2, 4, 3, 1, 1, 2, 4, 2, 8, 4, 3, 1, 1, 6, 4, 3, 2, 5, 4, 1, 1, 5, 2, 2, 3, 2, 2, 1, 1, 4, 5, 6, 2, 3, 5, 1, 1, 2, 3, 2, 4, 3, 6, 1, 1, 7, 8, 3, 2, 4, 5, 1, 1, 3, 2, 2, 3, 2, 2, 1, 1, 7, 3, 4, 2, 7, 6, 1, 1, 2, 5, 2, 5, 5, 3, 1, 1, 3, 3, 3, 2, 10, 3, 1, 1, 4, 2, 2

Offset: 2

N. J. A. Sloane and J. C. Lagarias, Sep 04 2002

nonn

We conjecture that an integer is always reached.
Is S(n) = Sum_{k=2..n} a(k) asymptotic to 3*n? S(n) = 3n for n = 69, 127, 166, 169, 189, 197, 327, 328, 360, 389, 404, 405, 419, 428, 497, 519, 520, 540, 541, 544, 547, 652, 668, 669, 676, 682, 683...
The sign of 3n-S(n) seems to change often: 3n-S(n) = 3, 4, 4, -1, 1, 3, 3, 4, 5, 5, 6, 7, 9, 11, 7, 6, 5, 6, 5, 5, 7, 9, 10, 9, 10, 5, 4, 4, 6, 8, 5, 4, 4, 5, 3, 2, 4, 6, 4, 5, 6, 6, 7, 8, 10, 12, 11, 9, 6, 7, 7, 5, 7, 9, 10, 10, 11, 10, 10, 7, 9, 11, 7, 2, 2, 3, 2, 0, 2, 4, 4, 5, 6, 6, 7, 8, 10, 12, 8, 8, 7, 8, 4, 1, 3, 5, 6, 4, 5, 3, 1, 1, 3, 5, 5, 5, 5, 6, -1, ... Is 3n-S(n) bounded? - Benoit Cloitre, Sep 05 2002

Robert Israel, Table of n, a(n) for n = 2..10000
J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.

Cf. A073524, A074735, A085785, A085817, A085833.

g := proc(x) local M,t1,t2,t3; M := 4^100; t1 := ceil(x) mod M; t2 := x*t1; t3 := numer(t2) mod M; t3/denom(t2); end;
a := []; for n from 2 to 150 do t1 := (2*n+1)/4; for i from 1 to 100 do t1 := g(t1); if type(t1,`integer`) then break; fi; od: a := [op(a),i]; od: a;

a[n_] := Length @ NestWhileList[# Ceiling[#]&, (2n+1)/4, !IntegerQ[#]&] - 1;
Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Jan 31 2023 *)

a(n)=if(n<1,0,s=n/2+1/4; c=0; while(frac(s)>0,s=s*ceil(s); c++); c) \\ Benoit Cloitre, Sep 05 2002

A081849 Consider recurrence b(0) = (2n+1)/2, b(n) = b(0)*ceiling(b(n-1)); sequence gives first integer reached.

Original entry on oeis.org

3, 20, 14, 468, 33, 299, 60, 47328, 95, 1218, 138, 25475, 189, 3161, 248, 20830128, 315, 6512, 390, 181138, 473, 11655, 564, 9015167, 663, 18974, 770, 671745, 885, 28853, 1008, 38906570560, 1139, 41676, 1278, 1799888, 1425, 57827, 1580, 110341278, 1743, 77690

Offset: 1

N. J. A. Sloane, Apr 13 2003

k = A001511(n) is the number of steps to reach an integer b(k).

Cf. A001511, A081853, A073524, A074078.

Digits := 100: c := ceil; A081849 := proc(a) local i,t0,t; t0 := a; t := 0; for i from 1 to 100 do if whattype(t0) <> integer then t0 := a*c(t0); t := t+1; else RETURN(t0); fi; od; RETURN('FAIL'); end;

a[n_]:=Module[{b=b0=(2n+1)/2},While[!IntegerQ[b],b=b0*Ceiling[b]]; b]; Array[a,42] (* Stefano Spezia, Jun 26 2024 *)

a(n) = if(n==1,3, my(t=2*n+1, k=1+valuation(n,2)); n*t^(k+1) >>k \ (t-2)); \\ Kevin Ryde, Jun 30 2024

from math import ceil
from fractions import Fraction
def a(n):
  b0 = b = Fraction((2*n+1), 2)
  while b.denominator != 1: b = b0*ceil(b)
  return b.numerator
print([a(n) for n in range(1, 43)]) # Michael S. Branicky, Mar 20 2021

a(n) = s*(n*s^k - 1/2) / (s-1) where s = b(0) = (2*n+1)/2 and k = A001511(n). - Kevin Ryde, Jun 30 2024

A075102 Number of steps to reach the first integer starting with 2^n/n and iterating the map x->x*ceiling(x), or -1 if no integer is ever reached.

Original entry on oeis.org

0, 0, 1, 0, 2, 5, 5, 0, 2, 4, 1, 1, 13, 12, 3, 0, 19, 3, 4, 8, 10, 11, 20, 1, 7, 1, 3, 8, 12, 1, 2, 0, 20, 6, 4, 7, 8, 23, 39, 1, 21, 8, 13, 16, 4, 1, 9, 7, 1, 6, 23, 30, 73, 6, 3, 14, 7, 7, 20, 12, 228, 16, 3, 0, 10, 5, 96, 3, 4, 13, 28, 3, 72, 57, 7, 9, 6, 6

Offset: 1

Reinhard Zumkeller, Sep 02 2002

nonn

The starting value is A075101(n)/A000265(n).

			a(5)=2 since (2^5)/5 = 32/5 -> 224/5 -> 2016 = A075103(5).

J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.

Cf. A000265, A068119, A073524, A075101, A075103.

More terms from Jinyuan Wang, Jan 15 2022

A081853 Consider recurrence b(0) = (2n+1)/2, b(n) = b(n-1)*ceiling(b(n-1)); sequence gives first integer reached.

Original entry on oeis.org

3, 60, 14, 268065, 33, 2093, 60, 1204154941925628, 95, 13398, 138, 701600900, 189, 47415, 248, 1489788110004539889867929328515560588293, 315, 123728, 390, 34427225343, 473, 268065, 564, 19873182780430314444725, 663, 512298, 770, 467193780498, 885, 894443, 1008

Offset: 1

N. J. A. Sloane, Apr 13 2003

nonn

J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.

A001511 gives number of steps to reach an integer. Cf. A081849, A073524, A074078.

Cf. A081854.

a[n_]:=Module[{b=(2n+1)/2},While[!IntegerQ[b],b*=Ceiling[b]]; b]; Array[a,31] (* Stefano Spezia, Jun 26 2024 *)

Define F(x) = x(x+1)/2. Write 2n+1 = 2^i*m + 2^(i-1) + 1, then a(n) = (1/2)F^(i-1)(2n+1). E.g. n=4, 2n+1 = 9 = 2^4*0 + 2^3 + 1, so i=4, m=0 and F(F(F(9))) = F(F(45)) = F(1035) = 536130, a(4) = 536130/2 = 268065.

a(30)-a(31) from Stefano Spezia, Jun 26 2024

A081850 Consider recurrence b(0) = (2n+1)/4, b(n) = b(0)*ceiling(b(n-1)); sequence gives number of steps to reach an integer (or -1 if no integer is ever reached).

Original entry on oeis.org

3, 2, 3, 13, 1, 1, 4, 2, 2, 9, 2, 2, 1, 1, 7, 3, 7, 3, 4, 3, 1, 1, 2, 4, 5, 10, 5, 13, 1, 1, 14, 8, 5, 2, 10, 11, 1, 1, 6, 2, 2, 17, 2, 2, 1, 1, 3, 6, 16, 5, 3, 4, 1, 1, 2, 4, 7, 9, 4, 3, 1, 1, 15, 9, 4, 2, 7, 5, 1, 1, 3, 2, 2, 3, 2, 2, 1, 1, 5, 5, 6, 5, 6, 4, 1, 1, 2, 4, 4, 3, 3, 11, 1, 1, 3, 3, 7, 2, 4

Offset: 2

N. J. A. Sloane, Apr 13 2003

nonn

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A073524, A081849, A081851.

Digits := 100: c := ceil; A081850 := proc(a) local i,t0,t; t0 := a; t := 0; for i from 1 to 100 do if whattype(t0) <> integer then t0 := a*c(t0); t := t+1; else RETURN(t); fi; od; RETURN('FAIL'); end;

Showing 1-10 of 23 results. Next

cp's OEIS Frontend

A073528 n at which A073524(n) sets a new record.

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A073529 Record values in A073524.

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A001511 The ruler function: exponent of the highest power of 2 dividing 2n. Equivalently, the 2-adic valuation of 2n.

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A068119 Number of steps to reach an integer starting with n + 1/4 and iterating the map x -> x*ceiling(x).

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A072340 Number of steps to reach an integer starting with n/3 and iterating the map x -> x*ceiling(x), or -1 if no integer is ever reached.

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A073341 Number of steps to reach an integer starting with (2n+1)/4 and iterating the map x -> x*ceiling(x).

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A081849 Consider recurrence b(0) = (2n+1)/2, b(n) = b(0)*ceiling(b(n-1)); sequence gives first integer reached.