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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)

x^(2^n) - a(n) is the minimal polynomial over Q for the algebraic number sqrt(p(1)*sqrt(p(2)*...*sqrt(p(n-1)*sqrt(p(n)))...)), where p(k) is the k-th prime. Each such monic polynomial is irreducible by Eisenstein's Criterion (using p = p(n)).

A prime version of Somos's quadratic recurrence sequence A052129(n) = A052129(n-1)^2 * n = Product_{k=1..n} k^(2^(n-k)). - Jonathan Sondow, Mar 29 2014

All positive integers have unique factorizations into powers of distinct primes, and into powers of squarefree numbers with distinct exponents that are powers of 2. (See A329332 for a description of the relationship between the two.) a(n) is the least number such that both factorizations have n factors. - Peter Munn, Dec 15 2019

From Peter Munn, Jan 24 2020 to Feb 06 2020: (Start)

For n >= 0, a(n+1) is the n-th power of 12 in the monoid defined by A306697.

a(n) is the least positive integer that cannot be expressed as the product of fewer than n terms of A072774 (powers of squarefree numbers).

All terms that are less than the order of the Monster simple group (A003131) are divisors of the group's order, with a(6) exceeding its square root.

(End)

It is remarkable that 4 of the first 5 terms are factorials. - Hal M. Switkay, Jan 21 2025

For all i, j:

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

Numbers k for which A001221(k) = A331591(k).

Numbers k that have the same number of terms in their factorization into powers of distinct primes as in their factorization into powers of squarefree numbers with distinct exponents that are powers of 2. See A329332 for a description of the relationship between the two factorizations and A225546.

If k is included, then all such x that A046523(x) = k are also included, i.e., all numbers with the same prime signature as k. Notably, primes (A000040) are included, but squarefree semiprimes (A006881) are not.

k^2 is included if and only if k is included, for example A001248 is included, but A085986 is not.

The ones in the binary expansion of n encode the Fermi-Dirac factors of a(n).

The following table gives the rank of the bit corresponding to the Fermi-Dirac factor p^2^k:

...

7| 9

5| 5 8

3| 2 4 7

2| 0 1 3 6

---+--------

p/k| 0 1 2 3 ...

This sequence is a bijection from the nonnegative integers to the positive integers with inverse A344536.

This sequence establishes a bijection from A261195 to A225547.

This sequence and A344535 each map between two useful choices for encoding sets of elements drawn from a 2-dimensional array. To give a very specific example, each mapping is an isomorphism between two alternative integer representations of the polynomial ring GF2[x,y]. The relevant set is {x^i*y^j : i, j >= 0}. The mappings between the two representations of the ring's addition operation are from XOR (A003987) to A059897(.,.) and for the multiplication operation, they are from A329331(.,.) to A329329(.,.). - Peter Munn, May 31 2021

The ones in the binary expansion of n encode the Fermi-Dirac factors of a(n).

The following table gives the rank of the bit corresponding to the Fermi-Dirac factor p^2^k:

...

7| 6

5| 3 7

3| 1 4 8

2| 0 2 5 9

---+--------

p/k| 0 1 2 3 ...

This sequence is a bijection from the nonnegative integers to the positive integers with inverse A344537.

This sequence establishes a bijection from A261195 to A225547.

These numbers are the "basic" fixed points of A225546.

This sequence is additive.

This sequence establishes a bijection from A225547 to A261195.

This sequence is additive.

This sequence establishes a bijection from A225547 to A261195.