cp's OEIS Frontend

This sequence and A001969 give the unique solution to the problem of splitting the nonnegative integers into two classes in such a way that sums of pairs of distinct elements from either class occur with the same multiplicities [Lambek and Moser]. Cf. A000028, A000379.

In French: les nombres impies.

Has asymptotic density 1/2, since exactly 2 of the 4 numbers 4k, 4k+1, 4k+2, 4k+3 have an even sum of bits, while the other 2 have an odd sum. - Jeffrey Shallit, Jun 04 2002

Nim-values for game of mock turtles played with n coins.

A115384(n) = number of odious numbers <= n; A000120(a(n)) = A132680(n). - Reinhard Zumkeller, Aug 26 2007

Indices of 1's in the Thue-Morse sequence A010060. - Tanya Khovanova, Dec 29 2008

For any positive integer m, the partition of the set of the first 2^m positive integers into evil ones E and odious ones O is a fair division for any polynomial sequence p(k) of degree less than m, that is, Sum_{k in E} p(k) = Sum_{k in O} p(k) holds for any polynomial p with deg(p) < m. - Pietro Majer, Mar 15 2009

For n>1 let b(n) = a(n-1). Then b(b(n)) = 2b(n). - Benoit Cloitre, Oct 07 2010

Lexicographically earliest sequence of distinct nonnegative integers with no term being the binary exclusive OR of any terms. The equivalent sequence for addition or for subtraction is A005408 (the odd numbers) and for multiplication is A026424. - Peter Munn, Jan 14 2018

Numbers of the form m XOR (2*m+1) for some m >= 0. - Rémy Sigrist, Apr 14 2022

This sequence and A000069 give the unique solution to the problem of splitting the nonnegative integers into two classes in such a way that sums of pairs of distinct elements from either class occur with the same multiplicities [Lambek and Moser]. Cf. A000028, A000379.

In French: les nombres païens.

Theorem: First differences give A036585. (Observed by Franklin T. Adams-Watters.)

Proof from Max Alekseyev, Aug 30 2006 (edited by N. J. A. Sloane, Jan 05 2021): (Start)

Observe that if the last bit of a(n) is deleted, we get the nonnegative numbers 0, 1, 2, 3, ... in order.

The last bit in a(n+1) is 1 iff the number of bits in n is odd, that is, iff A010060(n+1) is 1.

So, taking into account the different offsets here and in A010060, we have a(n) = 2*(n-1) + A010060(n-1).

Therefore the first differences of the present sequence equal 2 + first differences of A010060, which equals A036585. QED (End)

Integers k such that A010060(k-1)=0. - Benoit Cloitre, Nov 15 2003

Indices of zeros in the Thue-Morse sequence A010060 shifted by 1. - Tanya Khovanova, Feb 13 2009

Conjecture, checked up to 10^6: a(n) is also the sequence of numbers k representable as k = ror(x) XOR rol(x) (for some integer x) where ror(x)=A038572(x) is x rotated one binary place to the right, rol(x)=A006257(x) is x rotated one binary place to the left, and XOR is the binary exclusive-or operator. - Alex Ratushnyak, May 14 2016

From Charlie Neder, Oct 07 2018: (Start)

Conjecture is true: ror(x) and rol(x) have an even number of 1 bits in total (= 2 * A000120(x)), and XOR preserves the parity of this total, so the resulting number must have an even number of 1 bits. An x can be constructed corresponding to a(n) like so:

If the number of bits in a(n) is even, add a leading 0 so a(n) is 2k+1 bits long.

Do an inverse shuffle on a(n), then "divide" by 11, rotate the result k bits to the right, and shuffle to get x. (End)

Numbers of the form m XOR (2*m) for some m >= 0. - Rémy Sigrist, Feb 07 2021

The terms "evil numbers" and "odious numbers" were coined by Richard K. Guy, c. 1976 (Haque and Shallit, 2016) and appeared in the book by Berlekamp et al. (Vol. 1, 1st ed., 1982). - Amiram Eldar, Jun 08 2021

The original motivation for this sequence was to encode the prime factorization of n in the binary representation of a(n), each such representation being unique as long as this map is restricted to A005117 (squarefree numbers, resulting a permutation of nonnegative integers A048672) or any of its subsequence, resulting an injective function like A048623 and A048639.

However, also the restriction to A260443 (not all terms of which are squarefree) results a permutation of nonnegative integers, namely A001477, the identity permutation.

When a polynomial with nonnegative integer coefficients is encoded with the prime factorization of n (e.g., as in A206296, A260443), then a(n) gives the evaluation of that polynomial at x=2.

The primitive completely additive integer sequence that satisfies a(n) = a(A225546(n)), n >= 1. By primitive, we mean that if b is another such sequence, then there is an integer k such that b(n) = k * a(n) for all n >= 1. - Peter Munn, Feb 03 2020

If the binary rank of an integer partition y is given by Sum_i 2^(y_i-1), and the Heinz number is Product_i prime(y_i), then a(n) is the binary rank of the integer partition with Heinz number n. Note the function taking a set s to Sum_i 2^(s_i-1) is the inverse of A048793 (binary indices), and the function taking a multiset m to Product_i prime(m_i) is the inverse of A112798 (prime indices). - Gus Wiseman, May 22 2024

Inverse of sequence A006068 considered as a permutation of the nonnegative integers, i.e., A006068(A003188(n)) = n = A003188(A006068(n)). - Howard A. Landman, Sep 25 2001

Restricts to a permutation of each {2^(i - 1) .. 2^i - 1}. - Jason Kimberley, Apr 02 2012

a(n) mod 2 = floor(((n + 1) mod 4) / 2), see also A021913. - Reinhard Zumkeller, Apr 28 2012

Invented by Emile Baudot (1845-1903), originally called a "cyclic-permuted" code. Gray codes are named after Frank Gray, who patented their use for shaft encoders in 1953. [F. Gray, "Pulse Code Communication", U.S. Patent 2,632,058, March 17, 1953.] - Robert G. Wilson v, Jun 22 2014

For n >= 2, let G_n be the graph whose vertices are labeled as 0,1,...,2^n-1, and two vertices are adjacent if and only if their binary expansions differ in exactly one bit, then a(0),a(1),...,a(2^n-1),a(0) is a Hamilton cycle in G_n. - Jianing Song, Jun 01 2022

This sequence can be used for filtering certain base-2 related sequences, because it matches only with any such sequence b that can be computed as b(n) = f(A005940(n+1)), where f(n) is any function that depends only on the prime signature of n (some of these are listed under the index entry for "sequences computed from exponents in ...").

Matching in this context means that the sequence a matches with the sequence b iff for all i, j: a(i) = a(j) => b(i) = b(j). In other words, iff the sequence b partitions the natural numbers to the same or coarser equivalence classes (as/than the sequence a) by the distinct values it obtains.

Because the Doudna map n -> A005940(1+n) is an isomorphism from "unary-binary encoding of factorization" (see A156552) to the ordinary representation of the prime factorization of n, it follows that the equivalence classes of this sequence match with any such sequence b, where b(n) is computed from the lengths of 1-runs in the binary representation of n and the order of those 1-runs does not matter. Particularly, this holds for any sequence that is obtained as a "Run Length Transform", i.e., where b(n) = Product S(i), for some function S, where i runs through the lengths of runs of 1's in the binary expansion of n. See for example A227349.

However, this sequence itself is not a run length transform of any sequence (which can be seen for example from the fact that A046523 is not multiplicative).

Furthermore, this matches not only with sequences involving products of S(i), but with any sequence obtained with any commutative function applied cumulatively, like e.g., A000120 (binary weight, obtained in this case as Sum identity(i)), and A069010 (number of runs of 1's in binary representation of n, obtained as Sum signum(i)).

This is a self-inverse permutation of the nonnegative integers.

The function "substitute x+1 for x" on polynomials over GF(2) is completely multiplicative.

What is the density of fixed points in this sequence? Do we get a different answer if we look only at irreducible polynomials?

From Antti Karttunen, Dec 27 2013: (Start)

As what comes to the above question, the number of fixed points in range [2^(n-1),(2^n)-1] of the sequence is given by A131575(n). In range [0,0] there is one fixed point: 0, in range [1,1] there is also one: 1, in range [2,3] there are no fixed points, in range [4,7] there are two fixed points: 6 and 7, and so on. (Cf. also the C-code given in A118666.)

Similarly, the number of cycles in such ranges begins as 1, 1, 1, 3, 4, 10, 16, 36, 64, 136, ... which is A051437 shifted two steps right (prepended with 1's): Because the sequence is a self-inverse permutation, the number of its cycles in range [2^(n-1),(2^n)-1] is computed as: cycles(n) = (A011782(n)-number_of_fixedpoints(n))/2 + number_of_fixedpoints(n), which matches with the identity: A051437(n-2) = (A011782(n)-A131575(n))/2 + A131575(n), for n>=2.

In OEIS terms, the above comment about multiplicativeness can be rephrased as: a(A048720(x,y)) = A048720(a(x),a(y)) for all integers x, y >= 0. Here A048720(x,y) gives the product of carryless binary multiplication of x and y.

The permutation conjugates between Gray code and its inverse: A003188(n) = a(A006068(a(n))) and A006068(n) = a(A003188(a(n))) [cf. the identity 1.19-9d: gB = Bg^{-1} given on page 53 of fxtbook].

Because of the multiplicativity, the subset of irreducible (and respectively: composite) polynomials over GF(2) is closed under this permutation. Cf. the following mappings: a(A014580(n)) = A234750(n) and a(A091242(n)) = A234745(n).

(End)

Inverse of sequence A064358 considered as a permutation of the positive integers. - Howard A. Landman, Sep 25 2001

This sequence is not exactly a permutation because it has offset 0 but doesn't contain 0. A052331 is its exact inverse, which has offset 1 and contains 0. See also A064358.

Are there any other values of n besides 4 and 36 with a(n) = n? - Thomas Ordowski, Apr 01 2005

4 = 100 = 4^1 * 3^0 * 2^0, 36 = 100100 = 9^1 * 7^0 * 5^0 * 4^1 * 3^0 * 2^0. - Thomas Ordowski, May 26 2005

Ordering of positive integers by increasing "Fermi-Dirac representation", which is a representation of the "Fermi-Dirac factorization", term implying that each prime power with a power of two as exponent may appear at most once in the "Fermi-Dirac factorization" of n. (Cf. comment in A050376; see also the OEIS Wiki page.) - Daniel Forgues, Feb 11 2011

The subsequence consisting of the squarefree terms is A019565. - Peter Munn, Mar 28 2018

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. The FDH-number of a strict integer partition (y_1,...,y_k) is f(y_1)*...*f(y_k). A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. Then a(n) is the number whose binary indices are the parts of the strict integer partition with FDH-number n. - Gus Wiseman, Aug 19 2019

The set of indices of odd-valued terms has asymptotic density 0. In this sense (using the order they appear in this permutation) 100% of numbers are even. - Peter Munn, Aug 26 2019

a(n) = Product of distinct primorials larger than one, obtained as Product_{i} A002110(1+i), where i ranges over the zero-based positions of the 1-bits present in the binary representation of n.

This sequence can be represented as a binary tree. Each child to the left is obtained as A283980(k), and each child to the right is obtained as 2*A283980(k), when their parent contains k:

1

|

...................2....................

6 12

30......../ \........60 180......../ \......360

/ \ / \ / \ / \

/ \ / \ / \ / \

/ \ / \ / \ / \

210 420 1260 2520 6300 12600 37800 75600

etc.

This permutation is induced by a wreath recursion a = s(a,b), b = (b,b) (i.e., binary transducer, where s means that the bits at that state are toggled: 0 <-> 1) given on page 103 of the Bondarenko, Grigorchuk, et al. paper, starting from the active (swapping) state a and rewriting bits from the second most significant bit to the least significant end, continuing complementing as long as the first 1-bit is reached, which is the last bit to be complemented.

The automorphism group of infinite binary tree (isomorphic to an infinitely iterated wreath product of cyclic groups of two elements) embeds naturally into the group of "size-preserving Catalan bijections". Scheme-function psi gives an isomorphism that maps this kind of permutation to the corresponding Catalan automorphism/bijection (that acts on S-expressions). The following identities hold: *A069770 = psi(A063946) (just swap the left and right subtrees of the root), *A057163 = psi(A054429) (reflect the whole tree), *A069767 = psi(A153141), *A069768 = psi(A153142), *A122353 = psi(A006068), *A122354 = psi(A003188), *A122301 = psi(A154435), *A122302 = psi(A154436) and from *A154449 = psi(A154439) up to *A154458 = psi(A154448). See also comments at A153246 and A153830.

a(1) to a(2^n) is the sequence of row sequency numbers in a Hadamard-Walsh matrix of order 2^n, when constructed to give "dyadic" or Payley sequency ordering. - Ross Drewe, Mar 15 2014

In the Stern-Brocot enumeration system for positive rationals (A007305/A047679), this permutation converts the denominator into the numerator: A007305(n) = A047679(a(n)). - Yosu Yurramendi, Aug 01 2020

The binary digits of a(n) encode the prime factorization of A007913(n), where the i-th digit from the right is 1 if and only if prime(i) divides A007913(n), otherwise 0. - Robert Israel, Jan 12 2015

Old name: a(1) = 0; a(A000040(n)) = 2^(n-1), and a(n*m) = a(n) XOR a(m).

XOR is the bitwise exclusive or operation (A003987).

a(k^2) = 0 for a natural number k.

Equivalently, the i-th binary digit from the right is 1 iff prime(i) divides n an odd number of times, otherwise zero. - Ethan Beihl, Oct 15 2016

When a polynomial with nonnegative integer coefficients is encoded with the prime factorization of n (e.g., as in A206296, A260443, with scheme explained in A206284), then A048675(n) gives the evaluation of that polynomial at x=2. This sequence is otherwise similar, except the polynomial is evaluated over the field GF(2), which implies also that all its coefficients are essentially reduced modulo 2. - Antti Karttunen, Dec 11 2015

Squarefree numbers (A005117) give the positions k where a(k) = A048675(k). - Antti Karttunen, Oct 29 2016

From Peter Munn, Jun 07 2021: (Start)

When we encode polynomials with nonnegative integer coefficients as described by Antti Karttunen above, polynomial addition is represented by integer multiplication, multiplication is represented by A297845(.,.), and this sequence represents a surjective semiring homomorphism to polynomials in GF(2)[x] (encoded as described in A048720). The mapping of addition operations by this homomorphism is part of the sequence definition: "a(n*m) = a(n) XOR a(m)". The mapping of multiplication is given by a(A297845(n, k)) = A048720(a(n), a(k)).

In a related way, A329329 defines a representation of a different set of polynomials as positive integers, namely polynomials in GF(2)[x,y].

Let P_n(x,y) denote the polynomial represented, as in A329329, by n >= 1. If 0 is substituted for y in P_n(x,y), we get a polynomial P'_n(x,y) (in which y does not appear, of course) that is equivalent to a polynomial P'_n(x) in GF(2)[x]. a(n) is the integer encoding of P'_n(x) (described in A048720).

Viewed as above, this sequence represents another surjective homomorphism, a homomorphism between polynomial rings, with A329329(.,.)/A059897(.,.) and A048720(.,.)/A003987(.,.) as the respective ring operations.

a(n) can be composed as a(n) = A048675(A007913(n)) and the effect of the A007913(.) component corresponds to different operations on the respective polynomial domains of the two homomorphisms described above. In the first homomorphism, coefficients are reduced modulo 2; in the second, 0 is substituted for y. This is illustrated in the examples.

(End)