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A371731 Heinz numbers of non-biquanimous integer partitions. Numbers without a divisor having the same sum of prime indices as the quotient.

%I A371731 #9 Apr 07 2024 22:00:21
%S A371731 2,3,5,6,7,8,10,11,13,14,15,17,18,19,20,21,22,23,24,26,27,28,29,31,32,
%T A371731 33,34,35,37,38,39,41,42,43,44,45,46,47,50,51,52,53,54,55,56,57,58,59,
%U A371731 60,61,62,65,66,67,68,69,71,72,73,74,75,76,77,78,79,80
%N A371731 Heinz numbers of non-biquanimous integer partitions. Numbers without a divisor having the same sum of prime indices as the quotient.
%C A371731 These partitions are counted by A371795, even case A006827.
%C A371731 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%C A371731 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A371731 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.
%F A371731 Numbers n without a divisor d|n such that A056239(d) = A056239(n/d).
%e A371731 The prime indices of 975 are {2,3,3,6}, which are not biquanimous, so 975 is in the sequence.
%e A371731 The prime indices of 900 are {1,1,2,2,3,3}, which can be partitioned into {{1,2,3},{1,2,3}} or {{3,3},{1,1,2,2}}, so 900 is not in the sequence.
%t A371731 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A371731 biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
%t A371731 Select[Range[100],Not@*biqQ@*prix]
%Y A371731 The complement is A357976, counted by A002219.
%Y A371731 For prime signature instead of indices we have A371782, complement A371781.
%Y A371731 Partitions of this type are counted by A371795, even case A006827.
%Y A371731 A108917 counts knapsack partitions, ranks A299702, strict A275972.
%Y A371731 A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
%Y A371731 A237258 aerated counts biquanimous strict partitions, ranks A357854.
%Y A371731 A321142 and A371794 count non-biquanimous strict partitions.
%Y A371731 A321451 counts non-quanimous partitions, ranks A321453.
%Y A371731 A321452 counts quanimous partitions, ranks A321454.
%Y A371731 A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
%Y A371731 A371737 counts quanimous strict partitions, complement A371736.
%Y A371731 A371783 counts k-quanimous partitions.
%Y A371731 A371789 counts non-quanimous sets, differences A371790.
%Y A371731 A371791 counts biquanimous sets, differences A232466.
%Y A371731 A371792 counts non-biquanimous sets, differences A371793.
%Y A371731 A371796 counts quanimous sets, differences A371797.
%Y A371731 Cf. A000005, A002221, A027187, A035470, A064914, A299701, A300061, A318434, A321455, A326534, A357879, A366320, A367094.
%K A371731 nonn
%O A371731 1,1
%A A371731 _Gus Wiseman_, Apr 07 2024