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A106529
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Numbers having k prime factors (counted with multiplicity), the largest of which is the k-th prime.
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109
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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
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1,1
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COMMENTS
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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
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LINKS
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FORMULA
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EXAMPLE
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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.
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MAPLE
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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
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MATHEMATICA
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Select[Range@ 1560, PrimePi@ FactorInteger[#][[-1, 1]] == PrimeOmega@ # &] (* Michael De Vlieger, May 09 2015 *)
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CROSSREFS
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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.
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KEYWORD
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nonn
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AUTHOR
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Matthew Ryan (mattryan1994(AT)hotmail.com), May 30 2005
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STATUS
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approved
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