%I #5 May 14 2023 09:39:43

%S 10,14,20,22,26,28,30,33,34,38,39,40,42,44,46,50,51,52,56,57,58,60,62,

%T 66,68,69,70,74,76,78,80,82,84,85,86,87,88,90,92,93,94,95,98,99,100,

%U 102,104,106,110,111,112,114,115,116,117,118,120,122,123,124,126

%N Heinz numbers of 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.

%e The terms together with their prime indices begin:

%e 10: {1,3} 44: {1,1,5} 70: {1,3,4}

%e 14: {1,4} 46: {1,9} 74: {1,12}

%e 20: {1,1,3} 50: {1,3,3} 76: {1,1,8}

%e 22: {1,5} 51: {2,7} 78: {1,2,6}

%e 26: {1,6} 52: {1,1,6} 80: {1,1,1,1,3}

%e 28: {1,1,4} 56: {1,1,1,4} 82: {1,13}

%e 30: {1,2,3} 57: {2,8} 84: {1,1,2,4}

%e 33: {2,5} 58: {1,10} 85: {3,7}

%e 34: {1,7} 60: {1,1,2,3} 86: {1,14}

%e 38: {1,8} 62: {1,11} 87: {2,10}

%e 39: {2,6} 66: {1,2,5} 88: {1,1,1,5}

%e 40: {1,1,1,3} 68: {1,1,7} 90: {1,2,2,3}

%e 42: {1,2,4} 69: {2,9} 92: {1,1,9}

%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 A069900, complement A081306.

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

%Y Partitions of this type are counted by A237820.

%Y The complement is A362981, counted by A237824.

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

%K nonn

%O 1,1

%A _Gus Wiseman_, May 14 2023