A036554 -id:A036554 - cp's OEIS frontend

A007913 Squarefree part of n: a(n) is the smallest positive number m such that n/m is a square.

1, 2, 3, 1, 5, 6, 7, 2, 1, 10, 11, 3, 13, 14, 15, 1, 17, 2, 19, 5, 21, 22, 23, 6, 1, 26, 3, 7, 29, 30, 31, 2, 33, 34, 35, 1, 37, 38, 39, 10, 41, 42, 43, 11, 5, 46, 47, 3, 1, 2, 51, 13, 53, 6, 55, 14, 57, 58, 59, 15, 61, 62, 7, 1, 65, 66, 67, 17, 69, 70, 71, 2, 73, 74, 3, 19, 77

Offset: 1

R. Muller, Mar 15 1996

Also called core(n). [Not to be confused with the squarefree kernel of n, A007947.]
Sequence read mod 4 gives A065882. - Philippe Deléham, Mar 28 2004
This is an arithmetic function and is undefined if n <= 0.
A note on square roots of numbers: we can write sqrt(n) = b*sqrt(c) where c is squarefree. Then b = A000188(n) is the "inner square root" of n, c = A007913(n), lcm(A007947(b),c) = A007947(n) = "squarefree kernel" of n and bc = A019554(n) = "outer square root" of n. [Corrected by M. F. Hasler, Mar 01 2018]
If n > 1, the quantity f(n) = log(n/core(n))/log(n) satisfies 0 <= f(n) <= 1; f(n) = 0 when n is squarefree and f(n) = 1 when n is a perfect square. One can define n as being "epsilon-almost squarefree" if f(n) < epsilon. - Kurt Foster (drsardonicus(AT)earthlink.net), Jun 28 2008
a(n) is the smallest natural number m such that product of geometric mean of the divisors of n and geometric mean of the divisors of m are integers. Geometric mean of the divisors of number n is real number b(n) = Sqrt(n). a(n) = 1 for infinitely many n. a(n) = 1 for numbers from A000290: a(A000290(n)) = 1. For n = 8; b(8) = sqrt(8), a(n) = 2 because b(2) = sqrt(2); sqrt(8) * sqrt(2) = 4 (integer). - Jaroslav Krizek, Apr 26 2010
Dirichlet convolution of A010052 with the sequence of absolute values of A055615. - R. J. Mathar, Feb 11 2011
Booker, Hiary, & Keating outline a method for bounding (on the GRH) a(n) for large n using L-functions. - Charles R Greathouse IV, Feb 01 2013
According to the formula a(n) = n/A000188(n)^2, the scatterplot exhibits the straight lines y=x, y=x/4, y=x/9, ..., i.e., y=x/k^2 for all k=1,2,3,... - M. F. Hasler, May 08 2014
The Dirichlet inverse of this sequence is A008836(n) * A063659(n). - Álvar Ibeas, Mar 19 2015
a(n) = 1 if n is a square, a(n) = n if n is a product of distinct primes. - Zak Seidov, Jan 30 2016
All solutions of the Diophantine equation n*x=y^2 or, equivalently, G(n,x)=y, with G being the geometric mean, are of the form x=k^2*a(n), y=k*sqrt(n*a(n)), where k is a positive integer. - Stanislav Sykora, Feb 03 2016
If f is a multiplicative function then Sum_{d divides n} f(a(d)) is also multiplicative. For example, A010052(n) = Sum_{d divides n} mu(a(d)) and A046951(n) = Sum_{d divides n} mu(a(d)^2). - Peter Bala, Jan 24 2024

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 1000 terms from T. D. Noe)
Krassimir T. Atanassov, On Some of Smarandache's Problems.
Krassimir T. Atanassov, On the 22nd, 23rd, and the 24th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 2, 80-82.
Andrew Booker, Ghaith Hiary, and Jon Keating, Detecting squarefree numbers, CNTA XII (2012).
Henry Bottomley, Some Smarandache-type multiplicative sequences.
John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv preprint arXiv:1105.3399 [math.GM], 2011.
Vlad Copil and Laurenţiu Panaitopol, Properties of a sequence generated by positive integers, Bulletin mathématique de la Société des Sciences Mathématiques de Roumanie, Nouvelle Série, Vol. 50 (98), No. 2 (2007), pp. 131-137; alternative link.
Florentin Smarandache, Only Problems, Not Solutions!, Xiquan Publ., Phoenix-Chicago, 1993.
Eric Weisstein's World of Mathematics, Squarefree Part.

See A000188, A007947, A008833, A019554, A117811 for related information, specific to n.
See A027746, A027748, A124010 for factorization data for n.
Analogous sequences: A050985, A053165, A055231.
Cf. A002734, A005117 (range of values), A059897, A069891 (partial sums), A090699, A350389.
Related to A006519 via A225546.

a007913 n = product $
            zipWith (^) (a027748_row n) (map (`mod` 2) $ a124010_row n)
-- Reinhard Zumkeller, Jul 06 2012

[ Squarefree(n) : n in [1..256] ]; // N. J. A. Sloane, Dec 23 2006

A007913 := proc(n) local f,a,d; f := ifactors(n)[2] ; a := 1 ; for d in f do if type(op(2,d),'odd') then a := a*op(1,d) ; end if; end do: a; end proc: # R. J. Mathar, Mar 18 2011
# second Maple program:
a:= n-> mul(i[1]^irem(i[2], 2), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 20 2015
seq(n / expand(numtheory:-nthpow(n, 2)), n=1..77);  # Peter Luschny, Jul 12 2022

data = Table[Sqrt[n], {n, 1, 100}]; sp = data /. Sqrt[] -> 1; sfp = data/sp /. Sqrt[x] -> x (* Artur Jasinski, Nov 03 2008 *)
Table[Times@@Power@@@({#[[1]],Mod[ #[[2]],2]}&/@FactorInteger[n]),{n,100}] (* Zak Seidov, Apr 08 2009 *)
Table[{p, e} = Transpose[FactorInteger[n]]; Times @@ (p^Mod[e, 2]), {n, 100}] (* T. D. Noe, May 20 2013 *)
Sqrt[#] /. (c_:1)*a_^(b_:0) -> (c*a^b)^2& /@ Range@100 (* Bill Gosper, Jul 18 2015 *)

PARI
```
a(n)=core(n)
```

from sympy import factorint, prod
def A007913(n):
    return prod(p for p, e in factorint(n).items() if e % 2)
# Chai Wah Wu, Feb 03 2015

[squarefree_part(n) for n in (1..77)] # Peter Luschny, Feb 04 2015

Multiplicative with a(p^k) = p^(k mod 2). - David W. Wilson, Aug 01 2001
a(n) modulo 2 = A035263(n); a(A036554(n)) is even; a(A003159(n)) is odd. - Philippe Deléham, Mar 28 2004
Dirichlet g.f.: zeta(2s)*zeta(s-1)/zeta(2s-2). - R. J. Mathar, Feb 11 2011
a(n) = n/( Sum_{k=1..n} floor(k^2/n)-floor((k^2 -1)/n) )^2. - Anthony Browne, Jun 06 2016
a(n) = rad(n)/a(n/rad(n)), where rad = A007947. This recurrence relation together with a(1) = 1 generate the sequence. - Velin Yanev, Sep 19 2017
From Peter Munn, Nov 18 2019: (Start)
a(k*m) = A059897(a(k), a(m)).
a(n) = n / A008833(n).
(End)
a(A225546(n)) = A225546(A006519(n)). - Peter Munn, Jan 04 2020
From Amiram Eldar, Mar 14 2021: (Start)
Theorems proven by Copil and Panaitopol (2007):
Lim sup_{n->oo} a(n+1)-a(n) = oo.
Lim inf_{n->oo} a(n+1)-a(n) = -oo.
Sum_{k=1..n} 1/a(k) ~ c*sqrt(n) + O(log(n)), where c = zeta(3/2)/zeta(3) (A090699). (End)
a(n) = A019554(n)^2/n. - Jianing Song, May 08 2022
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/30 = 0.328986... . - Amiram Eldar, Oct 25 2022
a(n) = A007947(A350389(n)). - Amiram Eldar, Jan 20 2024

More terms from Michael Somos, Nov 24 2001

Definition reformulated by Daniel Forgues, Mar 24 2009

A276085 Primorial base log-function: fully additive with a(p) = p#/p, where p# = A034386(p).

Original entry on oeis.org

0, 1, 2, 2, 6, 3, 30, 3, 4, 7, 210, 4, 2310, 31, 8, 4, 30030, 5, 510510, 8, 32, 211, 9699690, 5, 12, 2311, 6, 32, 223092870, 9, 6469693230, 5, 212, 30031, 36, 6, 200560490130, 510511, 2312, 9, 7420738134810, 33, 304250263527210, 212, 10, 9699691, 13082761331670030, 6, 60, 13, 30032, 2312, 614889782588491410, 7, 216, 33, 510512, 223092871, 32589158477190044730, 10

Offset: 1

Antti Karttunen, Aug 21 2016

nonn

Completely additive with a(p^e) = e * A002110(A000720(p)-1).
This is a left inverse of A276086 ("primorial base exp-function"), hence the name "primorial base log-function". When the domain is restricted to the terms of A048103, this works also as a right inverse, as A276086(a(A048103(n))) = A048103(n) for all n >= 1. - Antti Karttunen, Apr 24 2022
On average, every third term is a multiple of 4. See A369001. - Antti Karttunen, May 26 2024

A left inverse of A276086.
Cf. A000040, A000720, A002110, A028234, A034386, A048103, A049345, A055396, A067029, A108951, A143293, A276154, A328316, A328624, A328625, A328768, A328832, A346105, A351576, A376398 (partial sums).
Positions of multiples of k in this sequence, for k=2, 3, 4, 5, 8, 27, 3125: A003159, A339746, A369002, A373140, A373138, A377872, A377878.
Cf. A036554 (positions of odd terms), A035263, A096268 (parity of terms).
Cf. A372575 (rgs-transform), A372576 [a(n) mod 360], A373842 [= A003415(a(n))].
Cf. A373145 [= gcd(A003415(n), a(n))], A373361 [= gcd(n, a(n))], A373362 [= gcd(A001414(n), a(n))], A373485 [= gcd(A083345(n), a(n))], A373835 [= gcd(bigomega(n), a(n))], and also A373367 and A373147 [= A003415(n) mod a(n)], A373148 [= a(n) mod A003415(n)].
Other completely additive sequences with primes p mapped to a function of p include: A001222 (with a(p)=1), A001414 (with a(p)=p), A059975 (with a(p)=p-1), A341885 (with a(p)=p*(p+1)/2), A373149 (with a(p)=prevprime(p)), A373158 (with a(p)=p#).
Cf. also A276075 for factorial base and A048675, A054841 for base-2 and base-10 analogs.

nn = 60; b = MixedRadix[Reverse@ Prime@ Range@ PrimePi[nn + 1]]; Table[FromDigits[#, b] &@ Reverse@ If[n == 1, {0}, Function[k, ReplacePart[Table[0, {PrimePi[k[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, k]]@ FactorInteger@ n], {n, nn}] (* Version 10.2, or *)
f[w_List] := Total[Times @@@ Transpose@ {Map[Times @@ # &, Prime@ Range@ Range[0, Length@ w - 1]], Reverse@ w}]; Table[f@ Reverse@ If[n == 1, {0}, Function[k, ReplacePart[Table[0, {PrimePi[k[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, k]]@ FactorInteger@ n], {n, 60}] (* Michael De Vlieger, Aug 30 2016 *)

A276085(n) = { my(f = factor(n), pr=1, i=1, s=0); for(k=1, #f~, while(i <= primepi(f[k, 1])-1, pr *= prime(i); i++); s += f[k, 2]*pr); (s); }; \\ Antti Karttunen, Nov 11 2024

from sympy import primorial, primepi, factorint
def a002110(n):
    return 1 if n<1 else primorial(n)
def a(n):
    f=factorint(n)
    return sum(f[i]*a002110(primepi(i) - 1) for i in f)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 22 2017

a(1) = 0; for n > 1, a(n) = a(A028234(n)) + (A067029(n) * A002110(A055396(n)-1)).
a(1) = 0, a(n) = (e1*A002110(i1-1) + ... + ez*A002110(iz-1)) when n = prime(i1)^e1 * ... * prime(iz)^ez.
Other identities.
For all n >= 0:
a(A276086(n)) = n.
a(A000040(1+n)) = A002110(n).
a(A002110(1+n)) = A143293(n).
From Antti Karttunen, Apr 24 & Apr 29 2022: (Start)
a(A283477(n)) = A283985(n).
a(A108951(n)) = A346105(n). [The latter has a similar additive formula as this sequence, but instead of primorials, uses their partial sums]
When applied to sequences where a certain subset of the divisors of n has been multiplicatively encoded with the help of A276086, this yields a corresponding number-theoretical sequence, i.e. completes their computation:
a(A319708(n)) = A001065(n) and a(A353564(n)) = A051953(n).
a(A329350(n)) = A069359(n) and a(A329380(n)) = A323599(n).
In the following group, the sum of the rhs-sequences is n [on each row, as say, A328841(n)+A328842(n)=n], because the pointwise product of the corresponding lhs-sequences is A276086:
a(A053669(n)) = A053589(n) and a(A324895(n)) = A276151(n).
a(A328571(n)) = A328841(n) and a(A328572(n)) = A328842(n).
a(A351231(n)) = A351233(n) and a(A327858(n)) = A351234(n).
a(A351251(n)) = A351253(n) and a(A324198(n)) = A351254(n).
The sum or difference of the rhs-sequences is A108951:
a(A344592(n)) = A346092(n) and a(A346091(n)) = A346093(n).
a(A346106(n)) = A346108(n) and a(A346107(n)) = A346109(n).
Here the two sequences are inverse permutations of each other:
a(A328624(n)) = A328625(n) and a(A328627(n)) = A328626(n).
a(A346102(n)) = A328622(n) and a(A346233(n)) = A328623(n).
a(A346101(n)) = A289234(n). [Self-inverse]
Other correspondences:
a(A324350(x,y)) = A324351(x,y).
a(A003961(A276086(n))) = A276154(n). [The primorial base left shift]
a(A276076(n)) = A351576(n). [Sequence reinterpreting factorial base representation as a primorial base representation]
(End)

Name amended by Antti Karttunen, Apr 24 2022

Name simplified, the old name moved to the comments - Antti Karttunen, Jun 23 2024

A003159 Numbers whose binary representation ends in an even number of zeros.

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 25, 27, 28, 29, 31, 33, 35, 36, 37, 39, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 89, 91, 92, 93, 95, 97, 99, 100, 101, 103, 105

Offset: 1

N. J. A. Sloane, Simon Plouffe

Fraenkel (2010) called these the "vile" numbers.
Minimal with respect to property that parity of number of 1's in binary expansion alternates.
Minimal with respect to property that sequence is half its complement. [Corrected by Aviezri S. Fraenkel, Jan 29 2010]
If k appears then 2k does not.
Increasing sequence of positive integers k such that A035263(k)=1 (from paper by Allouche et al.). - Emeric Deutsch, Jan 15 2003
a(n) is an odious number (see A000069) for n odd; a(n) is an evil number (see A001969) for n even. - Philippe Deléham, Mar 16 2004
Indices of odd numbers in A007913, in A001511. - Philippe Deléham, Mar 27 2004
Partial sums of A026465. - Paul Barry, Dec 09 2004
A121701(2*a(n)) = A121701(a(n)); A096268(a(n)-1) = 0. - Reinhard Zumkeller, Aug 16 2006
A different permutation of the same terms may be found in A141290 and the accompanying array. - Gary W. Adamson, Jun 14 2008
a(n) = n-th clockwise Tower of Hanoi move; counterclockwise if not in the sequence. - Gary W. Adamson, Jun 14 2008
Indices of terms of Thue-Morse sequence A010060 which are different from the previous term. - Tanya Khovanova, Jan 06 2009
The sequence has the following fractal property. Remove the odd numbers from the sequence, leaving 4,12,16,20,28,36,44,48,52,... Dividing these terms by 4 we get 1,3,4,5,7,9,11,12,..., which is the original sequence back again. - Benoit Cloitre, Apr 06 2010
From Gary W. Adamson, Mar 21 2010: (Start)
A conjectured identity relating to the partition sequence, A000041 as polcoeff p(x); A003159, and its characteristic function A035263: (1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, ...); and A036554 indicating n-th terms with zeros in A035263: (2, 6, 8, 10, 14, 18, 22, ...).
The conjecture states that p(x) = A(x) = A(x^2) when A(x) = polcoeff A174065 = the Euler transform of A035263 = 1/((1-x)*(1-x^3)*(1-x^4)*(1-x^5)*...) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + ... and the aerated variant = the Euler transform of the complement of A035263: 1/((1-x^2)*(1-x^6)*(1-x^8)*...) = 1 + x^2 + x^4 + 2*x^6 + 3*x^8 + 4*x^10 + ....
(End)
The conjecture above was proved by Jean-Paul Allouche on Dec 21 2013. - Gary W. Adamson, Jan 22 2014
If the lower s-Wythoff sequence of s is s, then s=A003159. (See A184117 for the definition of lower and upper s-Wythoff sequences.)  Starting with any nondecreasing sequence s of positive integers, A003159 is the limit when the lower s-Wythoff operation is iterated.  For example, starting with s=(1,4,9,16,...)=(n^2), we obtain lower and upper s-Wythoff sequences
  a=(1,3,4,5,6,8,9,10,11,12,14,...)=A184427;
  b=(2,7,12,21,31,44,58,74,...)=A184428.
  Then putting s=a and repeating the operation gives a'=(1,3,4,5,7,9,11,12,14,...), which has the same first eight terms as A003159. - Clark Kimberling, Jan 14 2011

			1=1, 3=11, 5=101 and 7=111 have no (0 = even) trailing zeros, 4=100 has 2 (= even) trailing zeros in the base-2 representation.
2=10 and 6=110 end in one (=odd number) of trailing zeros in their base-2 representation, therefore are not terms of this sequence. - _M. F. Hasler_, Oct 29 2013

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

Lars Blomberg, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Jean-Paul Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, arXiv preprint arXiv:1401.3727 [math.NT], 2014.
Jean-Paul Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, J. de Théorie des Nombres de Bordeaux, 27, no. 2 (2015), 375-388.
Jean-Paul Allouche, On the morphism 1 -> 121, 2 -> 12221, CNRS France, 2024. See pp. 3, 6.
Jean-Paul Allouche, On the morphism 1 -> 121, 2 -> 12221, Preprint, 2024 [Local copy, with permission]
Jean-Paul Allouche, Andre Arnold, Jean Berstel, Srecko Brlek, William Jockusch, Simon Plouffe, and Bruce E. Sagan, A sequence related to that of Thue-Morse, Discrete Math., 139 (1995), 455-461.
Jean-Paul Allouche and Jeffrey Shallit, The Ubiquitous Prouhet-Thue-Morse Sequence, in C. Ding. T. Helleseth and H. Niederreiter, eds., Sequences and Their Applications: Proceedings of SETA '98, Springer-Verlag, 1999, pp. 1-16.
Jean-Paul Allouche, Jeffrey Shallit, and Guentcho Skordev, Self-generating sets, integers with missing blocks and substitutions, Discrete Math. 292 (2005) 1-15.
George E. Andrews and David Newman, Binary Representations and Theta Functions, 2017.
L. Carlitz, R. Scoville, and V. E. Hoggatt, Jr., Representations for a special sequence, Fib. Quart., 10 (1972), 499-518, 550.
R. Clerico, P. Fabbri, and F. Ortenzio, Pericolosamente vicino a Collatz, Rudi Mathematici, N. 226 (Nov 2017), p. 14 (in Italian).
Michael Domaratzki, Trajectory-based codes, Acta Informatica, Volume 40, Numbers 6-7 / May, 2004.
Emeric Deutsch and Bruce E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, arXiv:math/0407326 [math.CO], 2004; J. Num. Theory 117 (2006), 191-215.
Artūras Dubickas, On the distance from a rational power to the nearest integer, Journal of Number Theory, Volume 117, Issue 1, March 2006, pp. 222-239.
Artūras Dubickas, On a sequence related to that of Thue-Morse and its applications, Discrete Math. 307 (2007), no. 9-10, 1082--1093. MR2292537 (2008b:11086).
Aviezri S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.
Aviezri S. Fraenkel, The vile, dopey, evil and odious game players, Discrete Math. 312 (2012), no. 1, 42-46.
Aviezri S. Fraenkel, Home Page.
Russell Jay Hendel, A Family of Sequences Generalizing the Thue Morse and Rudin Shapiro Sequences, arXiv:2505.20547 [cs.FL], 2025. See p. 3.
Clark Kimberling, Problem E2850, Amer. Math. Monthly, 87 (1980), 671.
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
Clark Kimberling, Affinely recursive sets and orderings of languages, Discrete Math., 274 (2004), 147-160.
N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, Slides of talk given at Rutgers University, Feb. 2012.
Jan Snellman, Greedy Regular Convolutions, arXiv:2504.02795 [math.NT], 2025.
Eric Sopena, i-Mark: A new subtraction division game, arXiv:1509.04199 [cs.DM], 2015.
David Wakeham and David R. Wood, On multiplicative Sidon sets, INTEGERS, 13 (2013), #A26.
Index entries for 2-automatic sequences.
Index entries for sequences related to binary expansion of n

For the actual binary numbers see A280049.
Indices of even numbers in A007814.
Complement of A036554, also one-half of A036554.
Cf. A001285, A010060, A000041, A174065, A292118.

import Data.List (delete)
a003159 n = a003159_list !! (n-1)
a003159_list = f [1..] where f (x:xs) = x : f (delete  (2*x) xs)
-- Reinhard Zumkeller, Nov 04 2011

filter:= n -> type(padic:-ordp(n,2),even):
select(filter,[$1..1000]); # Robert Israel, Jul 07 2014

f[n_Integer] := Block[{k = n, c = 0}, While[ EvenQ[k], c++; k /= 2]; c]; Select[ Range[105], EvenQ[ f[ # ]] & ]
Select[Range[150],EvenQ[IntegerExponent[#,2]]&] (* Harvey P. Dale, Oct 19 2011 *)

a(n)=if(n<2,n>0,n=a(n-1); until(valuation(n,2)%2==0,n++); n)

is(n)=valuation(n,2)%2==0 \\ Charles R Greathouse IV, Sep 23 2012

from itertools import count, islice
def A003159_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:(n&-n).bit_length()&1,count(max(startvalue,1)))
A003159_list = list(islice(A003159_gen(),30)) # Chai Wah Wu, Jul 11 2022

def A003159(n):
    def f(x):
        c, s = n+x, bin(x)[2:]
        l = len(s)
        for i in range(l&1^1,l,2):
            c -= int(s[i])+int('0'+s[:i],2)
        return c
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Jan 29 2025

a(0) = 1; for n >= 0, a(n+1) = a(n) + 1 if (a(n) + 1)/2 is not already in the sequence, = a(n) + 2 otherwise.
Limit_{n->oo} a(n)/n = 3/2. - Benoit Cloitre, Jun 13 2002
More precisely, a(n) = 3*n/2 + O(log n). - Charles R Greathouse IV, Sep 23 2012
a(n) = Sum_{k = 1..n} A026465(k). - Benoit Cloitre, May 31 2003
a(n+1) = (if a(n) mod 4 = 3 then A007814(a(n) + 1) mod 2 else a(n) mod 2) + a(n) + 1; a(1) = 1. - Reinhard Zumkeller, Aug 03 2003
a(A003157(n)) is even. - Philippe Deléham, Feb 22 2004
Sequence consists of numbers of the form 4^i*(2*j + 1), i>=0, j>=0. - Jon Perry, Jun 06 2004
G.f.: (1/(1-x)) * Product_{k >= 1} (1 + x^A001045(k)). - Paul Barry, Dec 09 2004
a(1) = 1, a(2) = 3, and for n >= 2 we get a(n+1) = 4 + a(n) + a(n-1) - a(a(n)-n+1) - a(a(n-1)-n+2). - Benoit Cloitre, Apr 08 2010
If A(x) is the counting function for a(n) <= x, then A(2^n) = (2^(n+1) + (-1)^n)/3. - Vladimir Shevelev, Apr 15 2010
a(n) = A121539(n) + 1. - Reinhard Zumkeller, Mar 01 2012
A003159 = { N | A007814(N) is even }. - M. F. Hasler, Oct 29 2013

Additional comments from Michael Somos
Edited by M. F. Hasler, Oct 29 2013
Incorrect formula removed by Peter Munn, Dec 04 2020

A225546 Tek's flip: Write n as the product of distinct factors of the form prime(i)^(2^(j-1)) with i and j integers, and replace each such factor with prime(j)^(2^(i-1)).

Original entry on oeis.org

1, 2, 4, 3, 16, 8, 256, 6, 9, 32, 65536, 12, 4294967296, 512, 64, 5, 18446744073709551616, 18, 340282366920938463463374607431768211456, 48, 1024, 131072, 115792089237316195423570985008687907853269984665640564039457584007913129639936, 24, 81, 8589934592, 36, 768

Offset: 1

Paul Tek, May 10 2013

This is a multiplicative self-inverse permutation of the integers.
A225547 gives the fixed points.
From Antti Karttunen and Peter Munn, Feb 02 2020: (Start)
This sequence operates on the Fermi-Dirac factors of a number. As arranged in array form, in A329050, this sequence reflects these factors about the main diagonal of the array, substituting A329050[j,i] for A329050[i,j], and this results in many relationships including significant homomorphisms.
This sequence provides a relationship between the operations of squaring and prime shift (A003961) because each successive column of the A329050 array is the square of the previous column, and each successive row is the prime shift of the previous row.
A329050 gives examples of how significant sets of numbers can be formed by choosing their factors in relation to rows and/or columns. This sequence therefore maps equivalent derived sets by exchanging rows and columns. Thus odd numbers are exchanged for squares, squarefree numbers for powers of 2 etc.
Alternative construction: For n > 1, form a vector v of length A299090(n), where each element v[i] for i=1..A299090(n) is a product of those distinct prime factors p(i) of n whose exponent e(i) has the bit (i-1) "on", or 1 (as an empty product) if no such exponents are present. a(n) is then Product_{i=1..A299090(n)} A000040(i)^A048675(v[i]). Note that because each element of vector v is squarefree, it means that each exponent A048675(v[i]) present in the product is a "submask" (not all necessarily proper) of the binary string A087207(n).
This permutation effects the following mappings:
A000035(a(n)) = A010052(n), A010052(a(n)) = A000035(n). [Odd numbers <-> Squares]
A008966(a(n)) = A209229(n), A209229(a(n)) = A008966(n). [Squarefree numbers <-> Powers of 2]
(End)
From Antti Karttunen, Jul 08 2020: (Start)
Moreover, we see also that this sequence maps between A016825 (Numbers of the form 4k+2) and A001105 (2*squares) as well as between A008586 (Multiples of 4) and A028983 (Numbers with even sum of the divisors).
(End)

			  7744  = prime(1)^2^(2-1)*prime(1)^2^(3-1)*prime(5)^2^(2-1).
a(7744) = prime(2)^2^(1-1)*prime(3)^2^(1-1)*prime(2)^2^(5-1) = 645700815.

Cf. A225547 (fixed points) and the subsequences listed there.
Transposes A329050, A329332.
An automorphism of positive integers under the binary operations A059895, A059896, A059897, A306697, A329329.
An automorphism of A059897 subgroups: A000379, A003159, A016754, A122132.
Permutes lists where membership is determined by number of Fermi-Dirac factors: A000028, A050376, A176525, A268388.
Also permutes: A036554, A059404, A066427, A066428, A072774, A331593.
Sequences f that satisfy f(a(n)) = f(n): A048675, A064179, A064547, A097248, A302777, A331592.
Pairs of sequences (f,g) that satisfy a(f(n)) = g(a(n)): (A000265,A008833), (A000290,A003961), (A005843,A334747), (A006519,A007913), (A008586,A334748).
Pairs of sequences (f,g) that satisfy a(f(n)) = g(n), possibly with offset change: (A000040,A001146), (A000079,A019565).
Pairs of sequences (f,g) that satisfy f(a(n)) = g(n), possibly with offset change: (A000035, A010052), (A008966, A209229), (A007814, A248663), (A061395, A299090), (A087207, A267116), (A225569, A227291).
Cf. A331287 [= gcd(a(n),n)].
Cf. A331288 [= min(a(n),n)], see also A331301.
Cf. A331309 [= A000005(a(n)), number of divisors].
Cf. A331590 [= a(a(n)*a(n))].
Cf. A331591 [= A001221(a(n)), number of distinct prime factors], see also A331593.
Cf. A331740 [= A001222(a(n)), number of prime factors with multiplicity].
Cf. A331733 [= A000203(a(n)), sum of divisors].
Cf. A331734 [= A033879(a(n)), deficiency].
Cf. A331735 [= A009194(a(n))].
Cf. A331736 [= A000265(a(n)) = a(A008833(n)), largest odd divisor].
Cf. A335914 [= A038040(a(n))].
A self-inverse isomorphism between pairs of A059897 subgroups: (A000079,A005117), (A000244,A062503), (A000290\{0},A005408), (A000302,A056911), (A000351,A113849 U {1}), (A000400,A062838), (A001651,A252895), (A003586,A046100), (A007310,A000583), (A011557,A113850 U {1}), (A028982,A042968), (A053165,A065331), (A262675,A268390).
A bijection between pairs of sets: (A001248,A011764), (A007283,A133466), (A016825, A001105), (A008586, A028983).
Cf. also A336321, A336322 (compositions with another involution, A122111).

Array[If[# == 1, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]] &, 28] (* Michael De Vlieger, Jan 21 2020 *)

A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));
a(n) = {my(f=factor(n)); for (i=1, #f~, my(p=f[i,1]); f[i,1] = A019565(f[i,2]); f[i,2] = 2^(primepi(p)-1);); factorback(f);} \\ Michel Marcus, Nov 29 2019

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A225546(n) = if(1==n,1,my(f=factor(n),u=#binary(vecmax(f[, 2])),prods=vector(u,x,1),m=1,e); for(i=1,u,for(k=1,#f~, if(bitand(f[k,2],m),prods[i] *= f[k,1])); m<<=1); prod(i=1,u,prime(i)^A048675(prods[i]))); \\ Antti Karttunen, Feb 02 2020

from math import prod
from sympy import prime, primepi, factorint
def A225546(n): return prod(prod(prime(i) for i, v in enumerate(bin(e)[:1:-1],1) if v == '1')**(1<Chai Wah Wu, Mar 17 2023

Multiplicative, with a(prime(i)^j) = A019565(j)^A000079(i-1).
a(prime(i)) = 2^(2^(i-1)).
From Antti Karttunen and Peter Munn, Feb 06 2020: (Start)
a(A329050(n,k)) = A329050(k,n).
a(A329332(n,k)) = A329332(k,n).
Equivalently, a(A019565(n)^k) = A019565(k)^n. If n = 1, this gives a(2^k) = A019565(k).
a(A059897(n,k)) = A059897(a(n), a(k)).
The previous formula implies a(n*k) = a(n) * a(k) if A059895(n,k) = 1.
a(A000040(n)) = A001146(n-1); a(A001146(n)) = A000040(n+1).
a(A000290(a(n))) = A003961(n); a(A003961(a(n))) = A000290(n) = n^2.
a(A000265(a(n))) = A008833(n); a(A008833(a(n))) = A000265(n).
a(A006519(a(n))) = A007913(n); a(A007913(a(n))) = A006519(n).
A007814(a(n)) = A248663(n); A248663(a(n)) = A007814(n).
A048675(a(n)) = A048675(n) and A048675(a(2^k * n)) = A048675(2^k * a(n)) = k + A048675(a(n)).
(End)
From Antti Karttunen and Peter Munn, Jul 08 2020: (Start)
For all n >= 1, a(2n) = A334747(a(n)).
In particular, for n = A003159(m), m >= 1, a(2n) = 2*a(n). [Note that A003159 includes all odd numbers]
(End)

Name edited by Peter Munn, Feb 14 2020

"Tek's flip" prepended to the name by Antti Karttunen, Jul 08 2020

A035263 Trajectory of 1 under the morphism 0 -> 11, 1 -> 10; parity of 2-adic valuation of 2n: a(n) = A000035(A001511(n)).

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1

Offset: 1

Karamanos Konstantinos, N. J. A. Sloane

First Feigenbaum symbolic (or period-doubling) sequence, corresponding to the accumulation point of the 2^{k} cycles through successive bifurcations.
To construct the sequence: start with 1 and concatenate: 1,1, then change the last term (1->0; 0->1) gives: 1,0. Concatenate those 2 terms: 1,0,1,0, change the last term: 1,0,1,1. Concatenate those 4 terms: 1,0,1,1,1,0,1,1 change the last term: 1,0,1,1,1,0,1,0, etc. - Benoit Cloitre, Dec 17 2002
Let T denote the present sequence. Here is another way to construct T. Start with the sequence S = 1,0,1,,1,0,1,,1,0,1,,1,0,1,,... and fill in the successive holes with the successive terms of the sequence T (from paper by Allouche et al.). - Emeric Deutsch, Jan 08 2003 [Note that if we fill in the holes with the terms of S itself, we get A141260. - N. J. A. Sloane, Jan 14 2009]
From N. J. A. Sloane, Feb 27 2009: (Start)
In more detail: define S to be 1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1,0,1___...
If we fill the holes with S we get A141260:
1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1, 0,
........1.........0.........1.........1.........0.......1.........1.........0...
- the result is
1..0..1.1.1..0..1.0.1..0..1.1.1..0..1.1.1..0..1.0.1.... = A141260.
But instead, if we define T recursively by filling the holes in S with the terms of T itself, we get A035263:
1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1, 0, 1___1, 0,
........1.........0.........1.........1.........1.......0.........1.........0...
- the result is
1..0..1.1.1..0..1.0.1..0..1.1.1..0..1.1.1..0..1.1.1.0.1.0.1..0..1.1.1..0..1.0.1.. = A035263. (End)
Characteristic function of A003159, i.e., A035263(n)=1 if n is in A003159 and A035263(n)=0 otherwise (from paper by Allouche et al.). - Emeric Deutsch, Jan 15 2003
This is the sequence of R (=1), L (=0) moves in the Towers of Hanoi puzzle: R, L, R, R, R, L, R, L, R, L, R, R, R, ... - Gary W. Adamson, Sep 21 2003
Manfred Schroeder, p. 279 states, "... the kneading sequences for unimodal maps in the binary notation, 0, 1, 0, 1, 1, 1, 0, 1..., are obtained from the Morse-Thue sequence by taking sums mod 2 of adjacent elements." On p. 278, in the chapter "Self-Similarity in the Logistic Parabola", he writes, "Is there a closer connection between the Morse-Thue sequence and the symbolic dynamics of the superstable orbits? There is indeed. To see this, let us replace R by 1 and C and L by 0." - Gary W. Adamson, Sep 21 2003
Partial sums modulo 2 of the sequence 1, a(1), a(1), a(2), a(2), a(3), a(3), a(4), a(4), a(5), a(5), a(6), a(6), ... . - Philippe Deléham, Jan 02 2004
Parity of A007913, A065882 and A065883. - Philippe Deléham, Mar 28 2004
The length of n-th run of 1's in this sequence is A080426(n). - Philippe Deléham, Apr 19 2004
Also parity of A005043, A005773, A026378, A104455, A117641. - Philippe Deléham, Apr 28 2007
Equals parity of the Towers of Hanoi, or ruler sequence (A001511), where the Towers of Hanoi sequence (1, 2, 1, 3, 1, 2, 1, 4, ...) denotes the disc moved, labeled (1, 2, 3, ...) starting from the top; and the parity of (1, 2, 1, 3, ...) denotes the direction of the move, CW or CCW. The frequency of CW moves converges to 2/3. - Gary W. Adamson, May 11 2007
A conjectured identity relating to the partition sequence, A000041: p(x) = A(x) * A(x^2) when A(x) = the Euler transform of A035263 = polcoeff A174065: (1 + x + x^2 + 2x^3 + 3x^4 + 4x^5 + ...). - Gary W. Adamson, Mar 21 2010
a(n) is 1 if the number of trailing zeros in the binary representation of n is even. - Ralf Stephan, Aug 22 2013
From Gary W. Adamson, Mar 25 2015: (Start)
A conjectured identity relating to the partition sequence, A000041 as polcoeff p(x); A003159, and its characteristic function A035263: (1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, ...); and A036554 indicating n-th terms with zeros in A035263: (2, 6, 8, 10, 14, 18, 22, ...).
The conjecture states that p(x) = A(x) = A(x^2) when A(x) = polcoeffA174065 = the Euler transform of A035263 = 1/(1-x)*(1-x^3)*(1-x^4)*(1-x^5)*... = (1 + x + x^2 + 2x^3 + 3x^4 + 4x^5 + ...) and the aerated variant = the Euler transform of the complement of A035263: 1/(1-x^2)*(1-x^6)*(1-x^8)*... = (1 + x^2 + x^4 + 2x^6 + 3x^8 + 4x^10 + ...).
(End)
The conjecture above was proved by Jean-Paul Allouche on Dec 21 2013.
Regarded as a column vector, this sequence is the product of A047999 (Sierpinski's gasket) regarded as an infinite lower triangular matrix and A036497 (the Fredholm-Rueppel sequence) where the 1's have alternating signs, 1, -1, 0, 1, 0, 0, 0, -1, .... - Gary W. Adamson, Jun 02 2021
The numbers of 1's through n (A050292) can be determined by starting with the binary (say for 19 = 1 0 0 1 1) and writing: next term is twice current term if 0, otherwise twice plus 1. The result is 1, 2, 4, 9, 19. Take the difference row, = 1, 1, 2, 5, 10; and add the odd-indexed terms from the right: 5, 4, 3, 2, 1 = 10 + 2 + 1 = 13. The algorithm is the basis for determining the disc configurations in the tower of Hanoi game, as shown in the Jul 24 2021 comment of A060572. - Gary W. Adamson, Jul 28 2021

Karamanos, Kostas. "From symbolic dynamics to a digital approach." International Journal of Bifurcation and Chaos 11.06 (2001): 1683-1694. (Full version. See p. 1685)
Karamanos, K. (2000). From symbolic dynamics to a digital approach: chaos and transcendence. In Michel Planat (Ed.), Noise, Oscillators and Algebraic Randomness (Lecture Notes in Physics, pp. 357-371). Springer, Berlin, Heidelberg. (Short version. See p. 359)
Manfred R. Schroeder, "Fractals, Chaos, Power Laws", W. H. Freeman, 1991
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 892, column 2, Note on p. 84, part (a).

Antti Karttunen, Table of n, a(n) for n = 1..65536 (first 10000 terms from T. D. Noe)
Ricardo Gómez Aíza, Symbolic dynamical scales: modes, orbitals, and transversals, arXiv:2009.02669 [math.DS], 2020.
Jean-Paul Allouche, On the morphism 1 -> 121, 2 -> 12221, CNRS France, 2024. See p. 6.
Jean-Paul Allouche, On the morphism 1 -> 121, 2 -> 12221, Preprint, 2024 [Local copy, with permission]
Jean-Paul Allouche, Andre Arnold, Jean Berstel, Srecko Brlek, William Jockusch, Simon Plouffe and Bruce E. Sagan, A sequence related to that of Thue-Morse, Discrete Math., 139 (1995), 455-461.
Jean-Paul Allouche and Michel Mendès France, Automata and Automatic Sequences, in: Axel F. and Gratias D. (eds), Beyond Quasicrystals. Centre de Physique des Houches, vol 3. Springer, Berlin, Heidelberg, pp. 293-367, 1995; DOI https://doi.org/10.1007/978-3-662-03130-8_11.
Jean-Paul Allouche and Michel Mendes France, Automata and Automatic Sequences, in: Axel F. and Gratias D. (eds), Beyond Quasicrystals. Centre de Physique des Houches, vol 3. Springer, Berlin, Heidelberg, pp. 293-367, 1995; DOI https://doi.org/10.1007/978-3-662-03130-8_11. [Local copy]
Jean-Paul Allouche and Jeffrey Shallit, The Ubiquitous Prouhet-Thue-Morse Sequence, in C. Ding. T. Helleseth and H. Niederreiter, eds., Sequences and Their Applications: Proceedings of SETA '98, Springer-Verlag, 1999, pp. 1-16.
Joerg Arndt, Matters Computational (The Fxtbook), pp.9-10, pp.734-738
Scott Balchin and Dan Rust, Computations for Symbolic Substitutions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.4.1.
Benoit Cloitre, Fractal walk on the 130000 first terms (step of unit length moving with angle Pi/3 if 0 and -Pi/3 if 1) starting at (0,0)
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
B. Derrida, A. Gervois and Y. Pomeau, Iteration of endomorphisms on the real axis and representation of number, Ann. Inst. Henri Poincaré, Section A: Physique Théorique, Vol. XXIX no. 3, 305-356 (1978).
Emeric Deutsch and Bruce E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, arXiv:math/0407326 [math.CO], 2004.
Emeric Deutsch and Bruce E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191-215.
Andreas M. Hinz, Sandi Klavžar, Uroš Milutinović, and Ciril Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 79. Book's website
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.
Konstantinos Karamanos, Symbolic Dynamics to a Digital Approach, Int. J. of Bifurcation and Chaos, 11(6), 1683-1694 (2001).
Konstantinos Karamanos and Grégoire Nicolis, Symbolic Dynamics and Entropy Analysis of Feigenbaum Limit Sets, Chaos, Solitons and Fractals 10(7), 1135 - 1150 (1999).
D. Kohel, S. Ling and C. Xing, Explicit Sequence Expansions
Joseph Meleshko, Pascal Ochem, Jeffrey Shallit, and Sonja Linghui Shan, Pseudoperiodic Words and a Question of Shevelev, arXiv:2207.10171 [math.CO], 2022.
Nicholas C. Metropolis, Myron L. Stein and Paul R. Stein, On finite limit sets for transformations on the unit interval, J. Combin. Theory, A 15 (1973), 25-44.
Jeffrey Shallit and Anatoly Zavyalov, Transduction of Automatic Sequences and Applications, arXiv:2303.15203 [cs.FL], 2023, see p. 5.
Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Eric Weisstein's World of Mathematics, Binary carry sequence.
Eric Weisstein's World of Mathematics, Double-free set.
Index entries for characteristic functions
Index entries for sequences related to binary expansion of n
Index entries for sequences that are fixed points of mappings

Parity of A001511. Anti-parity of A007814.
Absolute values of first differences of A010060. Apart from signs, same as A029883. Essentially the same as A056832.
Swapping 0 and 1 gives A096268.
Cf. A033485, A050292 (partial sums), A089608, A088172, A019300, A039982, A073675, A121701, A141260, A000041, A174065, A220466, A154269 (Mobius transform).
Limit of A317957(n) for large n.
Cf. A060572

import Data.Bits (xor)
a035263 n = a035263_list !! (n-1)
a035263_list = zipWith xor a010060_list $ tail a010060_list
-- Reinhard Zumkeller, Mar 01 2012

nmax:=105: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := (p+1) mod 2 od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Feb 07 2013
A035263 := n -> 1 - padic[ordp](n, 2) mod 2:
seq(A035263(n), n=1..105); # Peter Luschny, Oct 02 2018

a[n_] := a[n] = If[ EvenQ[n], 1 - a[n/2], 1]; Table[ a[n], {n, 1, 105}] (* Or *)
Rest[ CoefficientList[ Series[ Sum[ x^(2^k)/(1 + (-1)^k*x^(2^k)), {k, 0, 20}], {x, 0, 105}], x]]
f[1] := True; f[x_] := Xor[f[x - 1], f[Floor[x/2]]]; a[x_] := Boole[f[x]] (* Ben Branman, Oct 04 2010 *)
a[n_] := If[n == 0, 0, 1 - Mod[ IntegerExponent[n, 2], 2]]; (* Jean-François Alcover, Jul 19 2013, after Michael Somos *)
Nest[ Flatten[# /. {0 -> {1, 1}, 1 -> {1, 0}}] &, {0}, 7] (* Robert G. Wilson v, Jul 23 2014 *)
SubstitutionSystem[{0->{1,1},1->{1,0}},1,{7}][[1]] (* Harvey P. Dale, Jun 06 2022 *)

{a(n) = if( n==0, 0, 1 - valuation(n, 2)%2)}; /* Michael Somos, Sep 04 2006 */

{a(n) = if( n==0, 0, n = abs(n); subst( Pol(binary(n)) - Pol(binary(n-1)), x, 1)%2)}; /* Michael Somos, Sep 04 2006 */

{a(n) = if( n==0, 0, n = abs(n); direuler(p=2, n, 1 / (1 - X^((p<3) + 1)))[n])}; /* Michael Somos, Sep 04 2006 */

def A035263(n): return (n&-n).bit_length()&1 # Chai Wah Wu, Jan 09 2023

(define (A035263 n) (let loop ((n n) (i 1)) (cond ((odd? n) (modulo i 2)) (else (loop (/ n 2) (+ 1 i)))))) ;; (Use mod instead of modulo in R6RS) Antti Karttunen, Sep 11 2017

Absolute values of first differences (A029883) of Thue-Morse sequence (A001285 or A010060). Self-similar under 10->1 and 11->0.
Series expansion: (1/x) * Sum_{i>=0} (-1)^(i+1)*x^(2^i)/(x^(2^i)-1). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 17 2003
a(n) = Sum_{k>=0} (-1)^k*(floor((n+1)/2^k)-floor(n/2^k)). - Benoit Cloitre, Jun 03 2003
Another g.f.: Sum_{k>=0} x^(2^k)/(1+(-1)^k*x^(2^k)). - Ralf Stephan, Jun 13 2003
a(2*n) = 1-a(n), a(2*n+1) = 1. - Ralf Stephan, Jun 13 2003
a(n) = parity of A033485(n). - Philippe Deléham, Aug 13 2003
Equals A088172 mod 2, where A088172 = 1, 2, 3, 7, 13, 26, 53, 106, 211, 422, 845, ... (first differences of A019300). - Gary W. Adamson, Sep 21 2003
a(n) = a(n-1) - (-1)^n*a(floor(n/2)). - Benoit Cloitre, Dec 02 2003
a(1) = 1 and a(n) = abs(a(n-1) - a(floor(n/2))). - Benoit Cloitre, Dec 02 2003
a(n) = 1 - A096268(n+1); A050292 gives partial sums. - Reinhard Zumkeller, Aug 16 2006
Multiplicative with a(2^k) = 1 - (k mod 2), a(p^k) = 1, p > 2. Dirichlet g.f.: Product_{n = 4 or an odd prime} (1/(1-1/n^s)). - Christian G. Bower, May 18 2005
a(-n) = a(n). a(0)=0. - Michael Somos, Sep 04 2006
Dirichlet g.f.: zeta(s)*2^s/(2^s+1). - Ralf Stephan, Jun 17 2007
a(n+1) = a(n) XOR a(ceiling(n/2)), a(1) = 1. - Reinhard Zumkeller, Jun 11 2009
Let D(x) be the generating function, then D(x) + D(x^2) == x/(1-x). - Joerg Arndt, May 11 2010
a(n) = A010060(n) XOR A010060(n+1); a(A079523(n)) = 0; a(A121539(n)) = 1. - Reinhard Zumkeller, Mar 01 2012
a((2*n-1)*2^p) = (p+1) mod 2, p >= 0 and n >= 1. - Johannes W. Meijer, Feb 07 2013
a(n) = A000035(A001511(n)). - Omar E. Pol, Oct 29 2013
a(n) = 2-A056832(n) = (5-A089608(n))/4. - Antti Karttunen, Sep 11 2017, after Benoit Cloitre
For n >= 0, a(n+1) = M(2n) mod 2 where M(n) is the Motzkin number A001006 (see Deutsch and Sagan 2006 link). - David Callan, Oct 02 2018
a(n) = A038712(n) mod 3. - Kevin Ryde, Jul 11 2019
Given any n in the form (k * 2^m, k odd), extract k and m. Categorize the results into two outcomes of (k, m, even or odd). If (k, m) is (odd, even) substitute 1. If (odd, odd), denote the result 0.  Example: 5 = (5 * 2^0), (odd, even, = 1). (6 = 3 * 2^1), (odd, odd, = 0). - Gary W. Adamson, Jun 23 2021

Alternative description added to the name by Antti Karttunen, Sep 11 2017

A184117 Lower s-Wythoff sequence, where s(n) = 2n + 1.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 11, 12, 13, 15, 16, 18, 19, 20, 22, 23, 25, 26, 27, 29, 30, 32, 33, 35, 36, 37, 39, 40, 42, 43, 44, 46, 47, 49, 50, 52, 53, 54, 56, 57, 59, 60, 61, 63, 64, 66, 67, 69, 70, 71, 73, 74, 76, 77, 78, 80, 81, 83, 84, 85, 87, 88, 90, 91, 93, 94, 95, 97, 98, 100, 101, 102, 104, 105, 107, 108, 110, 111, 112, 114, 115, 117, 118, 119, 121, 122, 124, 125, 126, 128, 129, 131, 132, 134, 135, 136, 138, 139, 141

Offset: 1

Clark Kimberling, Jan 09 2011

nonn

Suppose that s(n) is a nondecreasing sequence of positive integers. The lower and upper s(n)-Wythoff sequences, a and b, are introduced here. Define
a(1) = 1; b(1) = s(1) + a(1); and for n>=2,
a(n) = least positive integer not in {a(1),...,a(n-1),b(1),...,b(n-1)},
b(n) = s(n) + a(n).
Clearly, a and b are complementary. If s(n)=n, then
a=A000201, the lower Wythoff sequence, and
b=A001950, the upper Wythoff sequence.
A184117 is chosen to represent the class of s-Wythoff sequences for which s is an arithmetic sequence given by s(n) = kn - r. Such sequences (lower and upper) are indexed in the OEIS as shown here:
n+1....A026273...A026274
n......A000201...A001950 (the classical Wythoff sequences)
2n+1...A184117...A184118
2n.....A001951...A001952
2n-1...A136119...A184119
3n+1...A184478...A184479
3n.....A184480...A001956
3n-1...A184482...A184483
3n-2...A184484...A184485
4n+1...A184486...A184487
4n.....A001961...A001962
4n-1...A184514...A184515
The pattern continues for A184516 to A184531.
s-Wythoff sequences for choices of s other than arithmetic sequences include these:
A184419 and A184420 (s = lower Wythoff sequence)
A184421 and A184422 (s = upper Wythoff sequence)
A184425 and A184426 (s = triangular numbers)
A184427 and A184428 (s = squares)
A036554 and A003159 (invariant and limiting sequences).

			s=(3,5,7,9,11,13,...);
a=(1,2,3,5,6,8,...);
b=(4,7,10,14,17,21,...).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Robbert Fokkink, Gerard Francis Ortega, and Dan Rust, Corner the Empress, arXiv:2204.11805 [math.CO], 2022. Mentions this sequence.

Cf. A000201, A001950, A001951, A001952, A003159, A036554.

k=2; r=-1;
mex:=First[Complement[Range[1,Max[#1]+1],#1]]&;
s[n_]:=k*n-r; a[1]=1; b[n_]:=b[n]=s[n]+a[n];
a[n_]:=a[n]=mex[Flatten[Table[{a[i],b[i]},{i,1,n-1}]]];
Table[s[n],{n,30}]  (* s = A005408 except for initial 1 *)
Table[a[n],{n,100}] (* a = A184117 *)
Table[b[n],{n,100}] (* b = A184118 *)

A184117_upto(N,s(n)=2*n+1,a=[1],U=a)={while(a[#a]1&&U[2]==U[1]+1,U=U[^1]);a=concat(a,U[1]+1));a} \\ M. F. Hasler, Jan 07 2019

a(n) = A184118(n) - s(n). - M. F. Hasler, Jan 07 2019

Removed an incorrect g.f., Alois P. Heinz, Dec 14 2012

A096268 Period-doubling sequence (or period-doubling word): fixed point of the morphism 0 -> 01, 1 -> 00.

Original entry on oeis.org

0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0

Offset: 0

N. J. A. Sloane, Jun 22 2004

nonn

Take highest power of 2 dividing n (A007814(n+1)), read modulo 2.
For the scale-invariance properties see Hendriks et al., 2012.
This is the sequence that results from the ternary Thue-Morse sequence (A036577) if all twos in that sequence are replaced by zeros. - Nathan Fox, Mar 12 2013
This sequence can be used to draw the Von Koch snowflake with a suitable walk in the plane. Start from the origin then the n-th step is "turn +Pi/3 if a(n)=0 and turn -2*Pi/3 if a(n)=1" (see link for a plot of the first 200000 steps). - Benoit Cloitre, Nov 10 2013
1 iff the number of trailing zeros in the binary representation of n+1 is odd. - Ralf Stephan, Nov 11 2013
Equivalently, with offset 1, the characteristic function of A036554 and an indicator for the A003159/A036554 classification of positive integers. - Peter Munn, Jun 02 2020

			Start: 0
Rules:
  0 --> 01
  1 --> 00
-------------
0:   (#=1)
  0
1:   (#=2)
  01
2:   (#=4)
  0100
3:   (#=8)
  01000101
4:   (#=16)
  0100010101000100
5:   (#=32)
  01000101010001000100010101000101
6:   (#=64)
  0100010101000100010001010100010101000101010001000100010101000100
7:   (#=128)
  010001010100010001000101010001010100010101000100010001010100010001000101010...
[_Joerg Arndt_, Jul 06 2011]

Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000 (first 1022 terms from T. D. Noe)
Jean-Paul Allouche, Michael Baake, Julien Cassaigne, and David Damanik, Palindrome complexity, arXiv:math/0106121 [math.CO], 2001; Theoretical Computer Science, 292 (2003), 9-31.
Scott Balchin and Dan Rust, Computations for Symbolic Substitutions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.4.1.
Benoit Cloitre, 200000 steps in the plane using "turn +Pi/3 if a(n)=0 and -2Pi/3 otherwise".
Cristian Cobeli, Mihai Prunescu, and Alexandru Zaharescu, On non-holonomicity, transcendence and p-adic valuations, arXiv:2412.16517 [math.NT], 2024. See p. 12.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
G. Jörg Endrullis, Dimitri Hendriks, and Jan Willem Klop, Degrees of streams.
Robbert Fokkink and Gandhar Joshi, Anti-recurrence sequences, arXiv:2506.13337 [math.NT], 2025. See pp. 2, 18.
Dimitri Hendriks, Frits G. W. Dannenberg, Jorg Endrullis, Mark Dow and Jan Willem Klop, Arithmetic Self-Similarity of Infinite Sequences, arXiv 1201.3786 [math.CO], 2012.
Andreas M. Hinz, Sandi Klavžar, Uroš Milutinović, and Ciril Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 79. Book's website
Shuo Li, Palindromic length sequence of the ruler sequence and of the period-doubling sequence, arXiv:2007.08317 [math.CO], 2020.
Laszlo Mérai and A. Winterhof, On the Nth linear complexity of automatic sequences, arXiv preprint arXiv:1711.10764 [math.NT], 2017.
Jeong-Yup Lee, D. Flom and S. I. Ben-Abraham, Multidimensional period doubling structures, Acta Crystallographica Section A: Foundations, (2016). A72, 391-394.
Aline Parreau, Michel Rigo, Eric Rowland, and Elise Vandomme, A new approach to the 2-regularity of the l-abelian complexity of 2-automatic sequences, arXiv:1405.3532 [cs.FL], 2014-2015.
Saúl Pilatowsky-Cameo, Soonwon Choi, and Wen Wei Ho, Critically slow Hilbert-space ergodicity in quantum morphic drives, arXiv:2502.06936 [quant-ph], 2025. See pp. 6, 15.
Narad Rampersad and Manon Stipulanti, The Formal Inverse of the Period-Doubling Sequence, arXiv:1807.11899 [math.CO], 2018.
Luke Schaeffer and Jeffrey Shallit, Closed, Palindromic, Rich, Privileged, Trapezoidal, and Balanced Words in Automatic Sequences, Electronic Journal of Combinatorics 23(1) (2016), Article P1.25.
Index entries for sequences that are fixed points of mappings.
Index entries for sequences related to binary expansion of n.

Not the same as A073059!
Swapping 0 and 1 gives A035263.
Cf. A096269, A096270, A071858, A220466, A036577.
Cf. A056832, A123087 (partial sums).
With offset 1, classification indicator for A003159/A036554.
Also with offset 1: A007814 mod 2 (cf. A096271 for mod 3), A048675 mod 2 (cf. A332813 for mod 3), A059975 mod 2.

a096268 = (subtract 1) . a056832 . (+ 1)
-- Reinhard Zumkeller, Jul 29 2014

[Valuation(n+1, 2) mod 2: n in [0..100]]; // Vincenzo Librandi, Jul 20 2016

nmax:=104: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 0 to ceil(nmax/(p+2))+1 do a((2*n+1)*2^p-1) := p mod 2 od: od: seq(a(n), n=0..nmax); # Johannes W. Meijer, Feb 02 2013
# second Maple program:
a:= proc(n) a(n):= `if`(n::even, 0, 1-a((n-1)/2)) end:
seq(a(n), n=0..125);  # Alois P. Heinz, Mar 20 2019

Nest[ Flatten[ # /. {0 -> {1, 0}, 1 -> {0, 0}}] &, {1}, 7] (* Robert G. Wilson v, Mar 05 2005 *)
{{0}}~Join~SubstitutionSystem[{0 -> {0, 1}, 1 -> {0, 0}}, {1}, 6] // Flatten (* Michael De Vlieger, Aug 15 2016 *)

a(n)=valuation(n+1,2)%2 \\ Ralf Stephan, Nov 11 2013

def A096268(n): return (~(n+1)&n).bit_length()&1 # Chai Wah Wu, Jan 09 2023

Recurrence: a(2*n) = 0, a(4*n+1) = 1, a(4*n+3) = a(n). - Ralf Stephan, Dec 11 2004
The recurrence may be extended backwards, with a(-1) = 1. - S. I. Ben-Abraham, Apr 01 2013
a(n) = 1 - A035263(n-1). - Reinhard Zumkeller, Aug 16 2006
Dirichlet g.f.: zeta(s)/(1+2^s). - Ralf Stephan, Jun 17 2007
Let T(x) be the g.f., then T(x) + T(x^2) = x^2/(1-x^2). - Joerg Arndt, May 11 2010
Let 2^k||n+1. Then a(n)=1 if k is odd, a(n)=0 if k is even. - Vladimir Shevelev, Aug 25 2010
a(n) = A007814(n+1) mod 2. - Robert G. Wilson v, Jan 18 2012
a((2*n+1)*2^p-1) = p mod 2, p >= 0 and n >= 0. - Johannes W. Meijer, Feb 02 2013
a(n) = A056832(n+1) - 1. - Reinhard Zumkeller, Jul 29 2014
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/3. = Amiram Eldar, Sep 18 2022

Corrected by Jeremy Gardiner, Dec 12 2004

More terms from Robert G. Wilson v, Feb 26 2005

A079523 Utterly odd numbers: numbers whose binary representation ends in an odd number of ones.

Original entry on oeis.org

1, 5, 7, 9, 13, 17, 21, 23, 25, 29, 31, 33, 37, 39, 41, 45, 49, 53, 55, 57, 61, 65, 69, 71, 73, 77, 81, 85, 87, 89, 93, 95, 97, 101, 103, 105, 109, 113, 117, 119, 121, 125, 127, 129, 133, 135, 137, 141, 145, 149, 151, 153, 157, 159, 161, 165, 167, 169, 173, 177, 181

Offset: 1

Benoit Cloitre, Jan 21 2003

Also, n such that A010060(n) = A010060(n+1) where A010060 is the Thue-Morse sequence.
Sequence of n such that a(n) = 3n begins 7, 23, 27, 29, 31, 39, 71, 87, 91, 93, 95, ...
Values of k such that the Motzkin number A001006(2k) is even. Values of k such that the number of restricted hexagonal polyominoes with 2k+1 cells is even (see A002212). Values of k such that the number of directed animals of size k+1 is even (see A005773). Values of k such that the Riordan number A005043(k) is even. - Emeric Deutsch and Bruce E. Sagan, Apr 02 2003
a(n) = A036554(n)-1 = A072939(n)-2. - Ralf Stephan, Jun 09 2003
Odious and evil terms alternate. - Vladimir Shevelev, Jun 22 2009
The sequence has the following fractal property: remove terms of the form 4k+1 from the sequence, and the remaining terms are of the form 4k+3: 7, 23, 31, 39, 55, 71, 87, ...; then subtract 3 from each of these terms and divide by 4 and you get the original sequence: 1, 5, 7, 9, 13, ... - Benoit Cloitre, Apr 06 2010
A035263(a(n)) = 0. - Reinhard Zumkeller, Mar 01 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J.-P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, arXiv preprint arXiv:1401.3727 [math.NT], 2014.
J.-P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, J. de Théorie des Nombres de Bordeaux, 27, no. 2 (2015), 375-388.
J.-P. Allouche, A. Arnold, J. Berstel, S. Brlek, W. Jockusch, Simon Plouffe, and B. E. Sagan, A relative of the Thue-Morse sequence, Discrete Math., 139, 1995, 455-461.
Narad Rampersad and Manon Stipulanti, The Formal Inverse of the Period-Doubling Sequence, arXiv:1807.11899 [math.CO], 2018.
Thomas Zaslavsky, Anti-Fibonacci Numbers: A Formula, Sep 26 2016 [Introduces the name "utterly odd". - _N. J. A. Sloane_, Sep 27 2016]
Index entries for 2-automatic sequences.

Cf. A003159, A003157, A003158, A075326, A131323.

import Data.List (elemIndices)
a079523 n = a079523_list !! (n-1)
a079523_list = elemIndices 0 a035263_list
-- Reinhard Zumkeller, Mar 01 2012

[n: n in [0..200] | Valuation(n+1, 2) mod 2 eq 0 + 1]; // Vincenzo Librandi, Apr 16 2015

Select[ Range[200], MatchQ[ IntegerDigits[#, 2], {b : (1) ..} | {_, 0, b : (1) ..} /; OddQ[ Length[{b}]]] & ] (* Jean-François Alcover, Jun 17 2013 *)

is(n)=valuation(n+1,2)%2 \\ Charles R Greathouse IV, Mar 07 2013

from itertools import count, islice
def A079523_gen(startvalue=1): return filter(lambda n:(~(n+1)&n).bit_length()&1,count(max(startvalue,1))) # generator of terms >= startvalue
A079523_list = list(islice(A079523_gen(),61)) # Chai Wah Wu, Jul 05 2022

def A079523(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, s = n+x, bin(x)[2:]
        l = len(s)
        for i in range(l&1,l,2):
            c -= int(s[i])+int('0'+s[:i],2)
        return c
    return bisection(f,n,n)-1 # Chai Wah Wu, Jan 29 2025

a(n) is asymptotic to 3n.
a(n) = 2*A003159(n) - 1. a(1)=1, a(n) = a(n-1) + 2 if (a(n-1)+1)/2 does not belong to the sequence and a(n) = a(n-1) + 4 otherwise. - Emeric Deutsch and Bruce E. Sagan, Apr 02 2003
a(n) = (1/2)*A081706(2n-1).
a(n) = A003158(n) - n = A003157(n) - n - 1. - Philippe Deléham, Feb 22 2004
Values of k such that A091297(k) = 0. - Philippe Deléham, Feb 25 2004

A033485 a(n) = a(n-1) + a(floor(n/2)), a(1) = 1.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 13, 18, 23, 30, 37, 47, 57, 70, 83, 101, 119, 142, 165, 195, 225, 262, 299, 346, 393, 450, 507, 577, 647, 730, 813, 914, 1015, 1134, 1253, 1395, 1537, 1702, 1867, 2062, 2257, 2482, 2707, 2969, 3231, 3530, 3829, 4175, 4521, 4914, 5307, 5757

Offset: 1

N. J. A. Sloane. This was in the 1973 "Handbook", but was then dropped from the database. Resubmitted by Philippe Deléham. Entry revised by N. J. A. Sloane, Jun 10 2012

Sequence gives the number of partitions of 2n into "strongly decreasing" parts (see the function s*(n) in the paper by Bessenrodt, Olsson, and Sellers); see the example in A040039.
a(A036554(n)) is even, a(A003159(n)) is odd. - Benoit Cloitre, Oct 23 2002
Partial sums of the sequence a(1)=1, a(1), a(1), a(2), a(2), a(3), a(3), a(4), a(4), a(5), a(5), a(6), ...; example: a(1) = 1, a(2) = 1+1 = 2, a(3) = 1+1+1 = 3, a(4) = 1+1+1+2 = 5, a(5) = 1+1+1+2+2 = 7, ... - Philippe Deléham, Jan 02 2004
The number of odd numbers before the n-th even number in this sequence is A003156(n). - Philippe Deléham, Mar 27 2004
There are no terms divisible by 4 and there are infinitely many terms divisible by {2,3,5,6,7,9,10,11,13,14,15} (see van Doorn link). - Ivan N. Ianakiev, Aug 06 2022 and Wouter van Doorn, Sep 17 2024
a(n) = A001401(n), for 1..14. A001401(15) = 84. - Wolfdieter Lang, Jan 09 2023

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..1000
J. Arkin, Problem H-102: Gone but not forgotten, Fibonacci Quarterly, Vol. 9 (1971), page 135.
Christine Bessenrodt, Jorn B. Olsson, and James A. Sellers, Unique path partitions: Characterization and Congruences, arXiv:1107.1156 [math.CO], 2011-2012.
Philippe Deléham, Letter to N. J. A. Sloane, Apr 20 1998
William Gasarch, What is known about that sequence, Computational Complexity blog, Jul 20 2022.
Wouter van Doorn, On the congruence properties and growth rate of a recursively defined sequence, arXiv:2406.09532 [math.NT], 2024.

Cf. A040039 (first differences), A178855 (partial sums).
Also half of A000123 (with first term omitted).
Cf. A022907.
Cf. A036554, A003159, A003156.
Cf. A007413, A178855, A131205.

import Data.List (transpose)
a033485 n = a033485_list !! (n-1)
a033485_list = 1 : zipWith (+)
   a033485_list (concat $ transpose [a033485_list, a033485_list])
-- Reinhard Zumkeller, Nov 15 2012

[n le 1 select 1 else Self(n-1) + Self(Floor(n/2)) : n in [1..60]]; // Vincenzo Librandi, Nov 20 2015

a:= proc(n) option remember;
      `if`(n<2, n, a(n-1)+a(iquo(n, 2)))
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Dec 16 2019

b[1]=1; b[n_] := b[n]=Sum[b[k], {k, 1, n/2}]; Table[b[n], {n, 3, 105, 2}] (* Robert G. Wilson v, Apr 22 2001 *)

PARI
```
a(n)=if(n<2,1,a(floor(n/2))+a(n-1))
```

from itertools import islice
from collections import deque
def A033485_gen(): # generator of terms
    aqueue, f, b, a = deque([2]), True, 1, 2
    yield from (1, 2)
    while True:
        a += b
        yield a
        aqueue.append(a)
        if f: b = aqueue.popleft()
        f = not f
A033485_list = list(islice(A033485_gen(),40)) # Chai Wah Wu, Jun 07 2022

a(n) = A000123(n)/2, for n >= 1.
Conjecture: lim_{n->oo} a(2n)/a(n)*log(n)/n = c = 1.64.... and a(n)/A(n) is bounded where A(n)=1 if n is a power of 2, otherwise A(n) = sqrt(n)*Product_{kBenoit Cloitre, Jan 26 2003
G.f.: A(x) satisfies x + (1+x)*A(x^2) = (1-x)*A(x). a(n) modulo 2 = A035263(n). - Philippe Deléham, Feb 25 2004
G.f.: (1/2)*(((1-x)*Product_{n>=0} (1-x^(2^n)))^(-1)-1). a(n) modulo 4 = A007413(n). - Philippe Deléham, Feb 28 2004
Sum_{k=1..n} a(k) = (a(2n+1)-1)/2 = A178855(n). - Philippe Deléham, Mar 18 2004
a(2n-1) = A131205(n). - Jean-Paul Allouche, Aug 11 2021
There exists a function f(n) such that n^f(n) < a(n) < n^(f(n) + epsilon) for all epsilon > 0 and all large enough n. - Wouter van Doorn, Sep 17 2024

cp's OEIS Frontend

A194688 First differences of A036554 (numbers whose binary representation ends in an odd number of zeros).

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A007913 Squarefree part of n: a(n) is the smallest positive number m such that n/m is a square.

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A276085 Primorial base log-function: fully additive with a(p) = p#/p, where p# = A034386(p).

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A003159 Numbers whose binary representation ends in an even number of zeros.

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A225546 Tek's flip: Write n as the product of distinct factors of the form prime(i)^(2^(j-1)) with i and j integers, and replace each such factor with prime(j)^(2^(i-1)).

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A035263 Trajectory of 1 under the morphism 0 -> 11, 1 -> 10; parity of 2-adic valuation of 2n: a(n) = A000035(A001511(n)).

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