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A371781 Numbers with biquanimous prime signature.

%I A371781 #18 Jul 24 2024 17:56:09
%S A371781 1,6,10,14,15,21,22,26,33,34,35,36,38,39,46,51,55,57,58,60,62,65,69,
%T A371781 74,77,82,84,85,86,87,90,91,93,94,95,100,106,111,115,118,119,122,123,
%U A371781 126,129,132,133,134,140,141,142,143,145,146,150,155,156,158,159
%N A371781 Numbers with biquanimous prime signature.
%C A371781 First differs from A320911 in lacking 900.
%C A371781 First differs from A325259 in having 1 and lacking 120.
%C A371781 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.
%C A371781 Also numbers n with a unitary divisor d|n having exactly half as many prime factors as n, counting multiplicity.
%H A371781 Robert Israel, <a href="/A371781/b371781.txt">Table of n, a(n) for n = 1..10000</a>
%e A371781 The prime signature of 120 is (3,1,1), which is not biquanimous, so 120 is not in the sequence.
%p A371781 biquanimous:= proc(L) local s,x,i,P; option remember;
%p A371781   s:= convert(L,`+`); if s::odd then return false fi;
%p A371781   P:= mul(1+x^i,i=L);
%p A371781   coeff(P,x,s/2) > 0
%p A371781 end proc:
%p A371781 select(n -> biquanimous(ifactors(n)[2][..,2]), [$1..200]); # _Robert Israel_, Apr 22 2024
%t A371781 g[n_]:=Select[Divisors[n],GCD[#,n/#]==1&&PrimeOmega[#]==PrimeOmega[n/#]&];
%t A371781 Select[Range[100],g[#]!={}&]
%t A371781 (* second program: *)
%t A371781 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 *)
%Y A371781 A number's prime signature is given by A124010.
%Y A371781 For prime indices we have A357976, counted by A002219 aerated.
%Y A371781 The complement for prime indices is A371731, counted by A371795, A006827.
%Y A371781 The complement is A371782, counted by A371840.
%Y A371781 Partitions of this type are counted by A371839.
%Y A371781 A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
%Y A371781 A237258 aerated counts biquanimous strict partitions, ranks A357854.
%Y A371781 A321142 and A371794 count non-biquanimous strict partitions.
%Y A371781 A321451 counts non-quanimous partitions, ranks A321453.
%Y A371781 A321452 counts quanimous partitions, ranks A321454.
%Y A371781 A371783 counts k-quanimous partitions.
%Y A371781 A371791 counts biquanimous sets, complement A371792.
%Y A371781 Cf. A035470, A064914, A232466, A299702, A336137, A371737, A371796.
%Y A371781 Subsequence of A028260.
%K A371781 nonn
%O A371781 1,2
%A A371781 _Gus Wiseman_, Apr 09 2024