This is a multiplicative self-inverse permutation of the integers.
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:
(End)
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)
