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A362981 Heinz numbers of integer partitions such that 2*(least part) >= greatest part.

%I A362981 #6 May 14 2023 09:39:47
%S A362981 1,2,3,4,5,6,7,8,9,11,12,13,15,16,17,18,19,21,23,24,25,27,29,31,32,35,
%T A362981 36,37,41,43,45,47,48,49,53,54,55,59,61,63,64,65,67,71,72,73,75,77,79,
%U A362981 81,83,89,91,96,97,101,103,105,107,108,109,113,119,121,125
%N A362981 Heinz numbers of integer partitions such that 2*(least part) >= greatest part.
%C A362981 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 A362981 By conjugation, also Heinz numbers of partitions whose greatest part appears at a middle position, namely k/2, (k+1)/2, or (k+2)/2, where k is the number of parts. These partitions have ranks A362622.
%e A362981 The terms together with their prime indices begin:
%e A362981      1: {}         16: {1,1,1,1}      36: {1,1,2,2}
%e A362981      2: {1}        17: {7}            37: {12}
%e A362981      3: {2}        18: {1,2,2}        41: {13}
%e A362981      4: {1,1}      19: {8}            43: {14}
%e A362981      5: {3}        21: {2,4}          45: {2,2,3}
%e A362981      6: {1,2}      23: {9}            47: {15}
%e A362981      7: {4}        24: {1,1,1,2}      48: {1,1,1,1,2}
%e A362981      8: {1,1,1}    25: {3,3}          49: {4,4}
%e A362981      9: {2,2}      27: {2,2,2}        53: {16}
%e A362981     11: {5}        29: {10}           54: {1,2,2,2}
%e A362981     12: {1,1,2}    31: {11}           55: {3,5}
%e A362981     13: {6}        32: {1,1,1,1,1}    59: {17}
%e A362981     15: {2,3}      35: {3,4}          61: {18}
%t A362981 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A362981 Select[Range[100],2*Min@@prix[#]>=Max@@prix[#]&]
%Y A362981 For prime factors instead of indices we have A081306.
%Y A362981 Prime indices are listed by A112798, length A001222, sum A056239.
%Y A362981 The complement is A362982, counted by A237820.
%Y A362981 Partitions of this type are counted by A237824.
%Y A362981 Cf. A027746, A053263, A171979, A237821, A327473, A327476, A362616, A362619, A362621, A362622.
%K A362981 nonn
%O A362981 1,2
%A A362981 _Gus Wiseman_, May 14 2023