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Analog of A249482 in Fermi-Dirac arithmetic, based on distinct terms of A050376 as "primes". Surprisingly there is a giant contrast between this sequence and A249482.

For k >= 1,

in interval [a(2k-1), a(2k)], L(n) <= 0,

in interval [a(2k), a(2k+1)], L(n) >= 0.

In particular, for k=1, in interval [2, 34], L(n) <= 0.

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)

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)

Not the same as A064380.

With n having a unique factorization as A050376(i) * A050376(j) * ... * A050376(k), with i, j, ..., k all distinct, a(n) = (A050376(i)-1) * (A050376(j)-1) * ... * (A050376(k)-1). (Cf. the first formula). - Antti Karttunen, Jan 15 2019

This sequence and A000379 (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.

Contains (for example) 180, so is different from A123193. - Max Alekseyev, Sep 20 2007

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

From Vladimir Shevelev, Oct 28 2013: (Start)

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

Number m is in the sequence if and only if the number k = k(m) of terms of A050376 which divide m with odd maximal exponent is odd.

For example, if m = 96, then the maximal exponent of 2 that divides 96 is 5, for 3 it is 1, for 4 it is 2, for 16 it is 1. Thus k(96) = 3 and 96 is a term.

(End)

Positions of odd terms in A064547, A268386 and A293439. - Antti Karttunen, Nov 09 2017

Lexicographically earliest sequence of distinct nonnegative integers such that no term is the A059897 product of 2 terms. (A059897 can be considered as a multiplicative operator related to the Fermi-Dirac factorization of numbers described in A050376.) Specifying that the A059897 product be of 2 distinct terms leaves the sequence unchanged. The equivalent sequences using standard integer multiplication are A026416 (with the 2 terms specified as distinct) and A026424 (otherwise). - Peter Munn, Mar 16 2019

From Amiram Eldar, Oct 02 2024: (Start)

Numbers whose number of infinitary divisors (A037445) is not a square.

Numbers whose exponentially odious part (A367514) has an odd number of distinct prime factors, i.e., numbers k such that A092248(A367514(k)) = 1. (End)

Completely additive modulo 3.

The equivalent sequence modulo 2 is A096268 (with offset 1), which produces the {A003159, A036554} classification.

Let H be the multiplicative subgroup of the positive rational numbers generated by the products of two consecutive primes and the cubes of primes. a(n) indicates the coset of H containing n. a(n) = 0 if n is in H. a(n) = 1 if n is in 2H. a(n) = -1 if n is in (1/2)H.

The properties of this classification can usefully be compared to two well-studied classifications. With the {A026424, A028260} classes, multiplying a member of one class by a prime gives a member of the other class. With the {A000028, A000379} classes, adding a factor to the Fermi-Dirac factorization of a member of one class gives a member of the other class. So, if 4 is not a Fermi-Dirac factor of k, k and 4k will be in different classes of the {A000028, A000379} set; but k and 4k will be in the same class of the {A026424, A028260} set. For two numbers to necessarily be in different classes when they differ in either of the 2 ways described above, 3 classes are needed.

With the classes defined by this sequence, no two of k, 2k and 4k are in the same class. This is a consequence of the following stronger property: if k is any positive integer and m is a member of A050376 (often called Fermi-Dirac primes), then no two of k, k * m, k * m^2 are in the same class. Also, if p and q are consecutive primes, then k * p and k * q are in different classes.

Further properties are given in the sequences that list the classes: A332820, A332821, A332822.

The scaled imaginary part of the Eisenstein integer-valued function, f, defined in A353445. - Peter Munn, Apr 27 2022

First differs from A378489 (the intersection of A000028 and A028260) by the inclusion of 72. - Peter Munn, Jul 13 2024

For all i, j:

A305801(i) = A305801(j) => a(i) = a(j),

a(i) = a(j) => A064179(i) = A064179(j),

a(i) = a(j) => A064547(i) = A064547(j),

a(i) = a(j) => A302777(i) = A302777(j),

a(i) = a(j) => A331308(i) = A331308(j),

a(i) = a(j) => A331287(i) = A331287(j),

a(i) = a(j) => A331592(i) = A331592(j).