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First differs from A359411 at n = 128.

Numbers whose prime factorization contains only exponents that are odd evil numbers (A129771).

First differs from A365298 at n = 64.

First differs from A050985 at n = 32.

First differs from A356192 at n = 64.

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)

Numbers having only squarefree exponents in their canonical prime factorization.

According to the formula of Theorem 3 [Toth], the density of the exponentially squarefree numbers is 0.9559230158619... (A262276). - Peter J. C. Moses and Vladimir Shevelev, Sep 10 2015

From Vladimir Shevelev, Sep 24 2015: (Start)

A generalization. Let S be a finite or infinite increasing integer sequence s=s(n), s(0)=0.

Let us call a positive number N an exponentially S-number, if all exponents in its prime power factorization are in the sequence S.

Let {u(n)} be the characteristic function of S. Then, for the density h=h(S) of the exponentially S-numbers, we have the representations

h(S) = Product_{prime p} Sum_{j in S} (p-1)/p^(j+1) = Product_{p} (1 + Sum_{j>=1} (u(j) - u(j-1))/p^j). In particular, if S = {0,1}, then the exponentially S-numbers are squarefree numbers; if S consists of 0 and {2^k}A138302%20(see%20%5BShevelev%5D,%202007);%20if%20S%20consists%20of%200%20and%20squarefree%20numbers,%20then%20u(n)=%7Cmu(n)%7C,%20where%20mu(n)%20is%20the%20M%C3%B6bius%20function%20(A008683),%20we%20obtain%20the%20density%20h%20of%20the%20exponentially%20squarefree%20numbers%20(cf.%20Toth's%20link,%20Theorem%203);%20the%20calculation%20of%20h%20with%20a%20very%20high%20degree%20of%20accuracy%20belongs%20to%20_Juan%20Arias-de-Reyna">{k>=0}, then the exponentially S-numbers form A138302 (see [Shevelev], 2007); if S consists of 0 and squarefree numbers, then u(n)=|mu(n)|, where mu(n) is the Möbius function (A008683), we obtain the density h of the exponentially squarefree numbers (cf. Toth's link, Theorem 3); the calculation of h with a very high degree of accuracy belongs to _Juan Arias-de-Reyna (A262276). Note that if S contains 1, then h(S) >= 1/zeta(2) = 6/Pi^2; otherwise h(S) = 0. Indeed, in the latter case, the density of the sequence of exponentially S-numbers does not exceed the density of A001694, which equals 0. (End)

The term "exponentially squarefree number" was apparently coined by Subbarao (1972). - Amiram Eldar, May 28 2025

This sequence and A000028 (its complement) give the unique solution to the problem of splitting the positive integers into two classes in such a way that products of pairs of distinct elements from either class occur with the same multiplicities [Lambek and Moser]. Cf. A000069, A001969.

See A000028 for precise definition, Maple program, etc.

The sequence contains products of even number of distinct terms of A050376. - Vladimir Shevelev, May 04 2010

From Vladimir Shevelev, Oct 28 2013: (Start)

Numbers m such that the infinitary Möbius function (A064179) of m equals 1. (This follows from the definition of A064179.)

A number m is in the sequence iff the number k = k(m) of terms of A050376 that divide m with odd maximal exponent is even (see example).

(End)

Numbers k for which A064547(k) [or equally, A268386(k)] is even. Numbers k for which A010060(A268387(k)) = 0. - Antti Karttunen, Feb 09 2016

The sequence is closed under the commutative binary operation A059897(.,.). As integers are self-inverse under A059897(.,.), it therefore forms a subgroup of the positive integers considered as a group under A059897(.,.). Specifically (expanding on the comment above dated May 04 2010) it is the subgroup of even length words in A050376, which is the group's lexicographically earliest ordered minimal set of generators. A000028, the set of odd length words in A050376, is its complementary coset. - Peter Munn, Nov 01 2019

From Amiram Eldar, Oct 02 2024: (Start)

Numbers whose number of infinitary divisors (A037445) is a square.

Numbers whose exponentially odious part (A367514) has an even number of distinct prime factors, i.e., numbers k such that A092248(A367514(k)) = 0. (End)

A268385 maps each term of this sequence to a unique term of A268335, and vice versa.

The asymptotic density of this sequence is Product_{p prime} f(1/p) = 0.87686263163054480657..., where f(x) = 1 - x + (1 - (1-x) * Product_{k>=0} (1-x^(2^k)))/2. - Amiram Eldar, Oct 27 2023