A106529 Numbers having k prime factors (counted with multiplicity), the largest of which is the k-th prime.

2, 6, 9, 20, 30, 45, 50, 56, 75, 84, 125, 126, 140, 176, 189, 196, 210, 264, 294, 315, 350, 396, 416, 440, 441, 490, 525, 594, 616, 624, 660, 686, 735, 875, 891, 924, 936, 968, 990, 1029, 1040, 1088, 1100, 1225, 1386, 1404, 1452, 1456, 1485, 1540, 1560

Offset: 1

Author

Matthew Ryan (mattryan1994(AT)hotmail.com), May 30 2005

Keywords

• nonn

Comments

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

The number of terms that have the k-th prime as their largest prime factor is A000984(k), the k-th central binomial coefficient. E.g., 6 and 9 are the A000984(2)=2 terms in {a(n)} that have prime(2)=3 as their largest prime factor.

The sequence contains the positive integers m such that the rank of the partition B(m) = 0. For 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

Also Heinz numbers of balanced partitions, counted by A047993. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - Gus Wiseman, Feb 08 2021

Examples

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000984.

A001222 counts prime factors.

A056239 adds up prime indices.

A061395 selects maximum prime index.

A112798 lists the prime indices of each positive integer.

Other balance-related sequences:

- A010054 counts balanced strict partitions.

- A047993 counts balanced partitions.

- A090858 counts partitions of rank 1.

- A098124 counts balanced compositions.

- A340596 counts co-balanced factorizations.

- A340598 counts balanced set partitions.

- A340599 counts alt-balanced factorizations.

- A340600 counts unlabeled balanced multiset partitions.

- A340653 counts balanced factorizations.

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

Programs

Maple

with(numtheory): a := proc (n) options operator, arrow: pi(max(factorset(n)))-bigomega(n) end proc: A := {}: for i from 2 to 1600 do if a(i) = 0 then A := `union`(A, {i}) else  end if end do: A; # Emeric Deutsch, May 09 2015

Mathematica

Select[Range@ 1560, PrimePi@ FactorInteger[#][[-1, 1]] == PrimeOmega@ # &] (* Michael De Vlieger, May 09 2015 *)

Formula

For all terms, A001222(a(n)) = A061395(a(n)). - Gus Wiseman, Feb 08 2021