OFFSET
1,2
COMMENTS
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.
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
EXAMPLE
The prime signature of 120 is (3,1,1), which is not biquanimous, so 120 is not in the sequence.
MAPLE
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
MATHEMATICA
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; EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, e}], x][[1 + sum/2]] > 0]; q[1] = True; Select[Range[200], q] (* Amiram Eldar, Jul 24 2024 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 09 2024
STATUS
approved