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A384349 Heinz numbers of integer partitions with no proper way to choose disjoint strict partitions of each part.

%I A384349 #7 Jun 05 2025 09:54:29
%S A384349 1,2,3,4,6,8,9,10,12,14,15,16,18,20,24,27,28,30,32,36,40,42,44,45,48,
%T A384349 50,52,54,56,60,63,64,66,68,70,72,75,76,78,80,81,84,88,90,92,96,98,99,
%U A384349 100,104,105,108,110,112,116,117,120,124,125,126,128,132,135
%N A384349 Heinz numbers of integer partitions with no proper way to choose disjoint strict partitions of each part.
%C A384349 By "proper" we exclude the case of all singletons, which is disjoint when n is squarefree.
%C A384349 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 A384349 The prime indices of 102 are {1,2,7}, which has proper disjoint choice ((1),(2),(3,4)), so 102 is not in the sequence.
%e A384349 The terms together with their prime indices begin:
%e A384349      1: {}           27: {2,2,2}        63: {2,2,4}
%e A384349      2: {1}          28: {1,1,4}        64: {1,1,1,1,1,1}
%e A384349      3: {2}          30: {1,2,3}        66: {1,2,5}
%e A384349      4: {1,1}        32: {1,1,1,1,1}    68: {1,1,7}
%e A384349      6: {1,2}        36: {1,1,2,2}      70: {1,3,4}
%e A384349      8: {1,1,1}      40: {1,1,1,3}      72: {1,1,1,2,2}
%e A384349      9: {2,2}        42: {1,2,4}        75: {2,3,3}
%e A384349     10: {1,3}        44: {1,1,5}        76: {1,1,8}
%e A384349     12: {1,1,2}      45: {2,2,3}        78: {1,2,6}
%e A384349     14: {1,4}        48: {1,1,1,1,2}    80: {1,1,1,1,3}
%e A384349     15: {2,3}        50: {1,3,3}        81: {2,2,2,2}
%e A384349     16: {1,1,1,1}    52: {1,1,6}        84: {1,1,2,4}
%e A384349     18: {1,2,2}      54: {1,2,2,2}      88: {1,1,1,5}
%e A384349     20: {1,1,3}      56: {1,1,1,4}      90: {1,2,2,3}
%e A384349     24: {1,1,1,2}    60: {1,1,2,3}      92: {1,1,9}
%t A384349 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A384349 pofprop[y_]:=Select[DeleteCases[Join@@@Tuples[IntegerPartitions/@y],y],UnsameQ@@#&];
%t A384349 Select[Range[100],Length[pofprop[prix[#]]]==0&]
%Y A384349 The non-proper version appears to be A382912, counted by A383710.
%Y A384349 The non-proper complement appears to be A382913, counted by A383708.
%Y A384349 The complement is A384321, counted by A384317.
%Y A384349 These partitions are counted by A384348.
%Y A384349 These are the positions of 0 in A384389.
%Y A384349 The case of a unique proper choice is A384390, counted by A384319.
%Y A384349 A048767 is the Look-and-Say transform, fixed points A048768.
%Y A384349 A056239 adds up prime indices, row sums of A112798.
%Y A384349 A179009 counts maximally refined strict partitions, ranks A383707.
%Y A384349 A279790 and A279375 count ways to choose disjoint strict partitions of prime indices.
%Y A384349 Cf. A122111, A130091, A317142, A326080, A351294, A357982, A381454, A382525, A383706, A384320, A384322.
%K A384349 nonn
%O A384349 1,2
%A A384349 _Gus Wiseman_, Jun 03 2025