A106529 - OEIS

%I #32 Oct 04 2021 16:09:01

%S 2,6,9,20,30,45,50,56,75,84,125,126,140,176,189,196,210,264,294,315,

%T 350,396,416,440,441,490,525,594,616,624,660,686,735,875,891,924,936,

%U 968,990,1029,1040,1088,1100,1225,1386,1404,1452,1456,1485,1540,1560

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

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

%C 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.

%C 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

%C 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

%H Alois P. Heinz, <a href="/A106529/b106529.txt">Table of n, a(n) for n = 1..10000</a>

%F For all terms, A001222(a(n)) = A061395(a(n)). - _Gus Wiseman_, Feb 08 2021

%e 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.

%p 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

%t Select[Range@ 1560, PrimePi@ FactorInteger[#][[-1, 1]] == PrimeOmega@ # &] (* _Michael De Vlieger_, May 09 2015 *)

%Y Cf. A000984.

%Y A001222 counts prime factors.

%Y A056239 adds up prime indices.

%Y A061395 selects maximum prime index.

%Y A112798 lists the prime indices of each positive integer.

%Y Other balance-related sequences:

%Y - A010054 counts balanced strict partitions.

%Y - A047993 counts balanced partitions.

%Y - A090858 counts partitions of rank 1.

%Y - A098124 counts balanced compositions.

%Y - A340596 counts co-balanced factorizations.

%Y - A340598 counts balanced set partitions.

%Y - A340599 counts alt-balanced factorizations.

%Y - A340600 counts unlabeled balanced multiset partitions.

%Y - A340653 counts balanced factorizations.

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

%K nonn

%O 1,1

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