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Heinz numbers of integer partitions such that 2*(least part) >= greatest part.
3

%I #6 May 14 2023 09:39:47

%S 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 36,37,41,43,45,47,48,49,53,54,55,59,61,63,64,65,67,71,72,73,75,77,79,

%U 81,83,89,91,96,97,101,103,105,107,108,109,113,119,121,125

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

%C 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 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 The terms together with their prime indices begin:

%e 1: {} 16: {1,1,1,1} 36: {1,1,2,2}

%e 2: {1} 17: {7} 37: {12}

%e 3: {2} 18: {1,2,2} 41: {13}

%e 4: {1,1} 19: {8} 43: {14}

%e 5: {3} 21: {2,4} 45: {2,2,3}

%e 6: {1,2} 23: {9} 47: {15}

%e 7: {4} 24: {1,1,1,2} 48: {1,1,1,1,2}

%e 8: {1,1,1} 25: {3,3} 49: {4,4}

%e 9: {2,2} 27: {2,2,2} 53: {16}

%e 11: {5} 29: {10} 54: {1,2,2,2}

%e 12: {1,1,2} 31: {11} 55: {3,5}

%e 13: {6} 32: {1,1,1,1,1} 59: {17}

%e 15: {2,3} 35: {3,4} 61: {18}

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],2*Min@@prix[#]>=Max@@prix[#]&]

%Y For prime factors instead of indices we have A081306.

%Y Prime indices are listed by A112798, length A001222, sum A056239.

%Y The complement is A362982, counted by A237820.

%Y Partitions of this type are counted by A237824.

%Y Cf. A027746, A053263, A171979, A237821, A327473, A327476, A362616, A362619, A362621, A362622.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 14 2023