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A371781
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Numbers with biquanimous prime signature.
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17
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1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 60, 62, 65, 69, 74, 77, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 126, 129, 132, 133, 134, 140, 141, 142, 143, 145, 146, 150, 155, 156, 158, 159
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OFFSET
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1,2
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COMMENTS
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First differs from A320911 in lacking 900.
First differs from A325259 in having 1 and lacking 120.
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 with 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 not in the sequence.
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MAPLE
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biquanimous:= proc(L) local s, x, i, P; option remember;
s:= convert(L, `+`); if s::odd then return false fi;
P:= mul(1+x^i, i=L);
coeff(P, x, s/2) > 0
end proc:
select(n -> biquanimous(ifactors(n)[2][.., 2]), [$1..200]); # Robert Israel, Apr 22 2024
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MATHEMATICA
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g[n_]:=Select[Divisors[n], GCD[#, n/#]==1&&PrimeOmega[#]==PrimeOmega[n/#]&];
Select[Range[100], g[#]!={}&]
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CROSSREFS
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A number's prime signature is given by A124010.
Partitions of this type are counted by A371839.
A371783 counts k-quanimous partitions.
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KEYWORD
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nonn,changed
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AUTHOR
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STATUS
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approved
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