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Numbers m with the property that, for some k, m has having k prime factors (counted with multiplicity), the largest of which is the k-th prime.

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a(7)=50 because we can write 50=2*5*5 and 5 is , for k=3, the product of k primes, the largest of which is the 3rd prime, 50 has 3 k-th prime factors , and 50 is the 7th such number.

The sequence contains the positive integers m such that the rank of the partition B(m) = 0. For n m >= 2, B(m) is defined as the partition obtained by taking the prime decomposition of m and replacing each prime factor p with its index i (i.e., i-th prime = p); also B(1) = the empty partition. For example, B(350) = B(2*5^2*7) = [1,3,3,4]. B is a bijection between the positive integers and the set of all partitions. The rank of a partition P is the largest part of P minus the number of parts of P. - Emeric Deutsch, May 09 2015

It seems that the ratio between successive terms tends to 1 as n increases , meaning perhaps that most numbers are in this sequence.

Each term can be expressed as a product of k primes and the The number of terms that all have the k-th prime as their largest prime factor is given by sequence A000984, which lists (k), the k-th central binomial coefficients, starting 1,2,6,20,70coefficient. E.g., 6 and 9 are the two A000984(2)=2 terms in A106529 with {a(n)} that have prime(2)=3 as their largest prime factor.

Sequence The sequence contains the positive integers n m such that the rank of the partition B(nm) = 0. For n >= 2, B(nm) is defined as the partition obtained by taking the prime decomposition of n m and replacing each prime factor p by with its index i (i.e. , i-th prime = p); also B(1) = the empty partition. For example, B(350) = B(2*5^2*7) = [1,3,3,4]. B is a bijection between the positive integers and the set of all partitions. The rank of a partition P is the largest part of P minus the number of parts of P. - Emeric Deutsch, May 09 2015

List of numbers n Numbers m with the property that , for some k, n has the k-th prime as its highest factor and m has k prime factors up to and including (counted with multiplicity), the largest of which is the k-th prime.

It seems like that the ratio between successive terms tends to 1 as n increases meaning perhaps that most numbers are in this sequence.

Each term can be expressed as a product of k primes and the number of terms that all have the k-th prime as their factor is given by sequence A000984, which are lists the central binomial coefficients, starting 1,2,6,20,70. E.g. , 6 and 9 are the two terms in A106529 with 3 as their highest largest prime factor.

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Cf. A001055, A006141, A064174, A096401, A117409, A168659, A324522, A340654, A340655, A340656/, A340657.