cp's OEIS Frontend

2 is the only term not divisible by 4. All powers of 2 are present. Every term divisible by an odd square is divisible by 16, the first such being 144.

The defined factorization is unique. Every positive number is a product of at most one squarefree number (A005117), at most one square of a squarefree number (A062503), at most one 4th power of a squarefree number (A113849), at most one 8th power of a squarefree number, and so on.

Iteratively map m using A000188, until 1 is reached, as A000188^k(m), for some k >= 1. m is in the sequence if and only if the preceding number, A000188^(k-1)(m), is 2. k can be shown to be A299090(m).

Closed under squaring, but not closed under multiplication: 12 = 3^1 * 2^2 and 432 = 3^1 * 3^2 * 2^4 are in the sequence, but 12 * 432 = 5184 = 3^4 * 2^6 = 2^2 * 6^4 is not.

The asymptotic density of this sequence is Sum_{k>=0} (d(2^(k+1)) - d(2^k))/2^(2^(k+1)-1) = 0.21363357193921052068..., where d(k) = 2^(k-1)/((2^k-1)*zeta(k)) is the asymptotic density of odd k-free numbers for k >= 2, and d(1) = 0. - Amiram Eldar, Feb 10 2024

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.

The complement is {35, 37, 39, 45, 61, 65, ...}.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}. This sequence lists all MM-numbers of regular multiset multisystems, where regularity means all vertex-degrees are equal.

Equivalently, biquadratefree (4th-power-free) part of square part of n.

Multiplicative. The terms are squares of squarefree numbers (A062503).

Every positive integer n is the product of a unique subset S_n of the terms of A050376 (sometimes called Fermi-Dirac primes). a(n) is the product of the members of S_n that are squares of prime numbers (A001248).

Every missing number greater than 2 is a multiple of 4. Every power of 2 is missing. Every positive power of every squarefree number greater than 2 is present.

The defined factorization is unique. Every positive number is a product of at most one squarefree number (A005117), at most one square of a squarefree number (A062503), at most one 4th power of a squarefree number (A113849), at most one 8th power of a squarefree number, and so on.

Iteratively map m using A000188, until 1 is reached, as A000188^k(m), for some k >= 1. m is in the sequence if and only if the preceding number, A000188^(k-1)(m), is greater than 2. k can be shown to be A299090(m).

The asymptotic density of this sequence is 1 - Sum_{k>=0} (d(2^(k+1)) - d(2^k))/2^(2^(k+1)-1) = 0.78636642806078947931..., where d(k) = 2^(k-1)/((2^k-1)*zeta(k)) is the asymptotic density of odd k-free numbers for k >= 2, and d(1) = 0. - Amiram Eldar, Feb 10 2024

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

Numbers k such that A034444(k) = A005361(k).

Numbers whose squarefree kernel (A007947) and powerful part (A057521) have the same number of divisors (A000005).

If k and m are coprime terms, then k*m is also a term.

All the terms are exponentially 2^n-numbers (A138302).

The characteristic function of this sequence depends only on prime signature.

Numbers whose canonical prime factorization has exponents whose geometric mean is 2.

Equivalently, numbers of the form Product_{i=1..m} p_i^(2^k_i), where p_i are distinct primes, and Sum_{i=1..m} k_i = m (i.e., the exponents k_i have an arithmetic mean 1).

1 is the only squarefree (A005117) term.

Includes the squares of squarefree numbers (A062503), which are the powerful (A001694) terms of this sequence.

The squares of primes (A001248) are the only terms that are prime powers (A246655).

Numbers of the for m*p^(2^k), where m is squarefree, p is prime, gcd(m, p) = 1 and omega(m) = k - 1, are all terms. In particular, this sequence includes numbers of the form p^4*q, where p != q are primes (A178739), and numbers of the form p^8*q*r where p, q, and r are distinct primes (A179747).

The corresponding numbers of squarefree (or powerful) divisors are 1, 2, 2, 2, 4, 4, 2, 4, 4, 4, 2, 4, ... . The least term with 2^k squarefree divisors is A360903(k).

A coreful divisor d of a number n is a divisor with the same set of distinct prime factors as n (see A307958).

The positive terms of A361430.