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A384390 Heinz numbers of integer partitions with a unique proper way to choose disjoint strict partitions of each part.

%I A384390 #12 Jul 27 2025 18:03:21
%S A384390 5,7,21,22,26,33,35,39,102,114,130,154,165,170,190,195,231,238,255,285
%N A384390 Heinz numbers of integer partitions with a unique proper way to choose disjoint strict partitions of each part.
%C A384390 By "proper" we exclude the case of all singletons, which is disjoint in the strict case.
%C A384390 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 A384390 The strict partition (7,2,1) with Heinz number 102 can only be properly refined as ((4,3),(2),(1)), so 102 is in the sequence. The other refinement ((7),(2),(1)) is not proper.
%e A384390 The terms together with their prime indices begin:
%e A384390     5: {3}
%e A384390     7: {4}
%e A384390    21: {2,4}
%e A384390    22: {1,5}
%e A384390    26: {1,6}
%e A384390    33: {2,5}
%e A384390    35: {3,4}
%e A384390    39: {2,6}
%e A384390   102: {1,2,7}
%e A384390   114: {1,2,8}
%e A384390   130: {1,3,6}
%e A384390   154: {1,4,5}
%e A384390   165: {2,3,5}
%e A384390   170: {1,3,7}
%e A384390   190: {1,3,8}
%e A384390   195: {2,3,6}
%e A384390   231: {2,4,5}
%e A384390   238: {1,4,7}
%e A384390   255: {2,3,7}
%e A384390   285: {2,3,8}
%t A384390 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A384390 pofprop[y_]:=Select[DeleteCases[Join@@@Tuples[IntegerPartitions/@y],y],UnsameQ@@#&];
%t A384390 Select[Range[100],Length[pofprop[prix[#]]]==1&]
%Y A384390 The non-proper version is A383707, counted by A179009.
%Y A384390 Partitions of this type are counted by A384319, non-strict A384323 (ranks A384347).
%Y A384390 This is the unique case of A384321, counted by A384317.
%Y A384390 This is the case of a unique proper choice in A384322.
%Y A384390 The complement is A384349 \/ A384393.
%Y A384390 These are positions of 1 in A384389.
%Y A384390 A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
%Y A384390 A055396 gives least prime index, greatest A061395.
%Y A384390 A056239 adds up prime indices, row sums of A112798.
%Y A384390 A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
%Y A384390 A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
%Y A384390 A357982 counts strict partitions of each prime index, non-strict A299200.
%Y A384390 Cf. A382912, counted by A383710, odd case A383711.
%Y A384390 Cf. A382913, counted by A383708, odd case A383533.
%Y A384390 Cf. A098859, A122111, A130091, A279375, A279790, A317142, A351201, A381454, A384005, A384320.
%K A384390 nonn
%O A384390 1,1
%A A384390 _Gus Wiseman_, Jun 02 2025