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A371782
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Numbers with non-biquanimous prime signature.
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16
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2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 30, 31, 32, 37, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 101, 102
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OFFSET
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1,1
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COMMENTS
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A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 (aerated) and ranked by A357976.
Also numbers n without a unitary divisor d|n having exactly half as many prime factors as n, counting multiplicity.
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LINKS
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EXAMPLE
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The prime signature of 120 is (3,1,1), which is not biquanimous, so 120 is in the sequence.
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MATHEMATICA
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g[n_]:=Select[Divisors[n], GCD[#, n/#]==1&&PrimeOmega[#]==PrimeOmega[n/#]&];
Select[Range[100], g[#]=={}&]
(* second program: *)
q[n_] := Module[{e = FactorInteger[n][[;; , 2]], sum, x}, sum = Plus @@ e; OddQ[sum] || CoefficientList[Product[1 + x^i, {i, e}], x][[1 + sum/2]] == 0]; q[1] = False; Select[Range[120], q] (* Amiram Eldar, Jul 24 2024 *)
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CROSSREFS
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A number's prime signature is given by A124010.
The complement for prime indices is A357976, counted by A002219 aerated.
Partitions of this type are counted by A371840.
A237258 (aerated) counts biquanimous strict partitions, ranks A357854.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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