cp's OEIS Frontend

A371782 Numbers with non-biquanimous prime signature.

%I A371782 #12 Jul 24 2024 17:55:41
%S A371782 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,
%T A371782 40,41,42,43,44,45,47,48,49,50,52,53,54,56,59,61,63,64,66,67,68,70,71,
%U A371782 72,73,75,76,78,79,80,81,83,88,89,92,96,97,98,99,101,102
%N A371782 Numbers with non-biquanimous prime signature.
%C A371782 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 A371782 Also numbers n without a unitary divisor d|n having exactly half as many prime factors as n, counting multiplicity.
%H A371782 Amiram Eldar, <a href="/A371782/b371782.txt">Table of n, a(n) for n = 1..10000</a>
%e A371782 The prime signature of 120 is (3,1,1), which is not biquanimous, so 120 is in the sequence.
%t A371782 g[n_]:=Select[Divisors[n],GCD[#,n/#]==1&&PrimeOmega[#]==PrimeOmega[n/#]&];
%t A371782 Select[Range[100],g[#]=={}&]
%t A371782 (* second program: *)
%t A371782 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 *)
%Y A371782 A number's prime signature is given by A124010.
%Y A371782 The complement for prime indices is A357976, counted by A002219 aerated.
%Y A371782 For prime indices we have A371731, counted by A371795, even case A006827.
%Y A371782 The complement is A371781, counted by A371839.
%Y A371782 Partitions of this type are counted by A371840.
%Y A371782 A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
%Y A371782 A237258 (aerated) counts biquanimous strict partitions, ranks A357854.
%Y A371782 A321142 and A371794 count non-biquanimous strict partitions.
%Y A371782 A321451 counts non-quanimous partitions, ranks A321453.
%Y A371782 A321452 counts quanimous partitions, ranks A321454.
%Y A371782 A371792 counts non-biquanimous sets, complement A371791.
%Y A371782 Cf. A035470, A064914, A232466, A299729, A367094, A371736, A371783, A371789.
%Y A371782 Subsequence of A026424.
%K A371782 nonn
%O A371782 1,1
%A A371782 _Gus Wiseman_, Apr 09 2024