cp's OEIS Frontend

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

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 k whose prime factorization k = p_1^e_1 * ... * p_m^e_m contains no pair of exponents e_i and e_j (i and j distinct) whose base-2 representations have at least one shared digit-position in which both exponents have a 1-bit.

Equivalently, numbers k such that the factors in the (unique) factorization of k into powers of squarefree numbers with distinct exponents that are powers of two, are prime powers. For example, this factorization of 90 is 10^1 * 3^2, so 90 is not included, as 10 is not prime; whereas this factorization of 320 is 5^1 * 2^2 * 2^4, so 320 is included as 5 and 2 are both prime. - Peter Munn, Jan 16 2020

A225546 maps the set of terms 1:1 onto A138302. - Peter Munn, Jan 26 2020

Equivalently, numbers k for which A064547(k) = A331591(k). - Amiram Eldar, Dec 23 2023

See A329332 for a description of the relationship between the two factorizations. From this relationship we get the formula a(n) = min(A001221(n), A001221(A225546(n))).

The result depends only on the prime signature of n.

k first appears at A191555(k).

Numbers of the form s^(2^e), where s is a nonunit squarefree number, and e >= 0.

The categorization provided by this sequence and its complement, A340681, is an alternative extension (to all integers greater than 1) of the 2-way distinction between squarefree and nonsquarefree as it applies to nonsquares.

All positive integers have a unique factorization into powers of nonunit squarefree numbers with distinct exponents that are powers of 2. This sequence lists the numbers where this factorization has only one term, that is numbers m such that A331591(m) = 1.

Presence in the sequence is determined by prime signature. The set of represented signatures starts: {{1}, {2}, {1,1}, {1,1,1}, {4}, {2,2}, {1,1,1,1}, {1,1,1,1,1}, {2,2,2}, {1,1,1,1,1,1}, {1,1,1,1,1,1,1}, {8}, {4,4}, {2,2,2,2}, {1,1,1,1,1,1,1,1}, ...}. Representing each signature in the set by the least number with that signature, we get the set A133492.

Positions of terms > 1 in A340675.

This is well-defined because positive integers have a unique factorization into powers of nonunit squarefree numbers with distinct exponents that are powers of 2.

Each (infinite) row is the lexicographically earliest with product n and terms that are a (2^k)-th power for all k.

For all k, column k is column k+1 of A060176 conjugated by A225546.

Nonunit squarefree numbers take the value 2, other nonsquares take the value 1, and squares take the square of the value taken by their square root.

Original, equal definition: totally additive with a(p^e) = e * A005187(2^(PrimePi(p)-1)), where PrimePi(n) = A000720(n).