cp's OEIS Frontend

A squarefree analog of A207901 (and the subsequence consisting of its squarefree terms): Each term is either a divisor or a multiple of the next one, and the terms differ by a single prime factor. Compare also to A284003.

For all n >= 0, max(a(n + 1), a(n)) / min(a(n + 1), a(n)) = A094290(n + 1) = prime(valuation(n + 1, 2) + 1) = A000040(A001511(n + 1)). [See Russ Cox's Dec 04 2010 comment in A007814.] - David A. Corneth & Antti Karttunen, Apr 16 2018

A multiplicative analog of absolute difference A049581. Exponents in prime factorization of T(n,k) are absolute differences of those of n and k. Commutative non-associative operator with identity 1. T(nx,kx)=T(n,k), T(n^x,k^x)=T(n,k)^x, etc.

The bivariate function log(T(., .)) is a distance (or metric) function. It is a weighted analog of A130836, in the sense that if e_i (resp. f_i) denotes the exponent of prime p_i in the factorization of m (resp. of n), then both log(T(m, n)) and A130836(m, n) are writable as Sum_{i} w_i * abs(e_i - f_i). For A130836, w_i = 1 for all i, whereas for log(T(., .)), w_i = log(p_i). - Luc Rousseau, Sep 17 2018

If the analog of absolute difference, as described in the first comment, is determined by factorization into distinct terms of A050376 instead of by prime factorization, the equivalent operation is defined by A059897 and is associative. The positive integers form a group under A059897. The two factorization methods give the same factorization for squarefree numbers (A005117), so that T(.,.) restricted to A005117 is associative. Thus the squarefree numbers likewise form a group under the operation defined by this sequence. - Peter Munn, Apr 04 2019

Let m be an odd positive number. Let S_m denote the sequence {Product_{i=1..r} q_(n+t_i)}A050376%20and%20Sum">{n>=1}, where {q_i} is sequence A050376 and Sum{i=1..r} 2^(t_1 - t_i) is the binary representation of m, such that t_1 > t_2 > ... > t_r = 0. Note that {S_1, S_3, S_5, ...} is a partition of all integers > 1. Then S_1=A050376, which is obtained when we set r=1, t_1 = 0. [Formula made compatible with A240535 data by Peter Munn, Aug 10 2021]

This present sequence is S_3 in this partition. It is obtained when we set r=2, t_1=1, t_2=0.

S_m(n) = A052330(A030101(m)*2^(n-1)) = A329330(A050376(n), A052330(A030101(m))). - Peter Munn, Aug 10 2021

A minimal set of generators for A000379 as a group under A059897(.,.). - Peter Munn, Aug 11 2019

Every positive integer is the product of a unique subset of the terms of A050376 (sometimes called Fermi-Dirac primes). This sequence lists the numbers where the relevant subset includes 3 but not 2.

Numbers whose squarefree part is divisible by 3 but not 2.

Positive multiples of 3 that survive sieving by the rule: if m appears then 2m, 3m and 6m do not. Asymptotic density is 1/6.

Self-inverse permutation of the natural numbers.

The asymptotic density of this sequence is 1/2. - Amiram Eldar, Apr 03 2022

From Peter Munn, Nov 16 2023: (Start)

Contains all nonzero squares.

Dividing by 5 the terms that are multiples of 5 gives its complement, A225838.

(A352272, 2*A352272, 3*A352272, 6*A352272) is a partition of the terms.

The terms form a subgroup of the positive integers under the operation A059897(.,.) and are the positive integers in an index 2 multiplicative subgroup of rationals that is generated by 2, 3 and integers congruent to 1 modulo 6. See A225857 and A352272 for further information about such subgroups.

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For the lexicographic ordering, the prime factors must be written in nonincreasing order. If we write the factors in nondecreasing order, we get a lexicographically ordered set with an order type that is greater than a natural number index - the resulting sequence does not include all qualifying numbers. (Note also that the symbols used for the lexicographic order are the prime numbers, not their digits.)

a(n) is the n-th power of 4 in the monoid defined in A331590.

Conjecture: a(n) is the position of the first occurrence of n in A334109.

Numbers of the form 4^i * 9^j * (6k+1), i, j, k >= 0.

Closed under multiplication.

The sequence forms a subgroup of the positive integers under the commutative operation A059897(.,.), one of 8 subgroups of the form {k : A007913(k) == 1 (mod m)} - in each case m is a divisor of 24. A059897 has a relevance to squarefree parts that arises from the identity A007913(k*n) = A059897(A007913(k), A007913(n)), where A007913(n) is the squarefree part of n.

The subgroup has 8 cosets, which partition the positive integers as follows. For each i in {1, 5}, j in {1, 2, 3, 6} there is a coset {m^2 * (6k+i) * j : m >= 1, k >= 0}. See the table in the examples.

None of the 8 cosets have been entered into the database previously, but many subgroups of the quotient group (which are formed of certain combinations of cosets) are represented among earlier OEIS sequences, including 6 of the 7 subgroups of index 2 (which combine 4 cosets). This sequence can therefore be defined as the intersection of pairs or triples of these sequences in many combinations (see the cross-references). See also the table in the example section of A352273 (the coset that includes 5).

Every positive square (A000290 without 0) is the product of a unique subset of these numbers. The lexicographically earliest (when ordered) minimal set of generators for the positive squares as a group under A059897(.,.); the intersection of A050376 and A000290. - Peter Munn, Aug 25 2019

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

From Peter Munn, Feb 04 2022: (Start)

For every odd squarefree number, s, exactly one of s and 2s is a term.

Closed under the commutative operation A350066(.,.).

Closed under the commutative operation A059897(.,.) forming a subgroup of the positive integers considered as a group under A059897. As subgroups, this sequence and A028982 are each a transversal of the other.

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