

A106529


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


79



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
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OFFSET

1,1


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 kth prime as their largest prime factor is A000984(k), the kth 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., ith 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


LINKS

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


FORMULA

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


EXAMPLE

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


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 *)


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 balancerelated sequences:
 A010054 counts balanced strict partitions.
 A047993 counts balanced partitions.
 A090858 counts partitions of rank 1.
 A098124 counts balanced compositions.
 A340596 counts cobalanced factorizations.
 A340598 counts balanced set partitions.
 A340599 counts altbalanced factorizations.
 A340600 counts unlabeled balanced multiset partitions.
 A340653 counts balanced factorizations.
Cf. A006141, A064174, A096401, A117409, A168659, A324522, A340654, A340655, A340656, A340657.
Sequence in context: A301798 A093840 A129233 * A325040 A350949 A088902
Adjacent sequences: A106526 A106527 A106528 * A106530 A106531 A106532


KEYWORD

nonn


AUTHOR

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


STATUS

approved



