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A384347 Heinz numbers of integer partitions with exactly two possible ways to choose disjoint strict partitions of each part.

%I A384347 #7 May 28 2025 10:53:22
%S A384347 5,7,21,22,25,26,33,35,39,49,102,114,130,147,154,165,170,175,190,195,
%T A384347 231,238,242,255,275,285
%N A384347 Heinz numbers of integer partitions with exactly two possible ways to choose disjoint strict partitions of each part.
%C A384347 Positions of 2 in A383706.
%C A384347 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 A384347 The prime indices of 275 are {3,3,5}, with two ways to choose disjoint strict partitions of each part: ((3),(2,1),(5)) and ((2,1),(3),(5)). Hence 275 is in the sequence.
%e A384347 The terms together with their prime indices begin:
%e A384347     5: {3}
%e A384347     7: {4}
%e A384347    21: {2,4}
%e A384347    22: {1,5}
%e A384347    25: {3,3}
%e A384347    26: {1,6}
%e A384347    33: {2,5}
%e A384347    35: {3,4}
%e A384347    39: {2,6}
%e A384347    49: {4,4}
%e A384347   102: {1,2,7}
%e A384347   114: {1,2,8}
%e A384347   130: {1,3,6}
%e A384347   147: {2,4,4}
%e A384347   154: {1,4,5}
%e A384347   165: {2,3,5}
%t A384347 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A384347 pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y],UnsameQ@@#&];
%t A384347 Select[Range[100],Length[pof[prix[#]]]==2&]
%Y A384347 The case of no choices is A382912, counted by A383710, odd case A383711.
%Y A384347 These are positions of 2 in A383706.
%Y A384347 The case of no proper choices is A383707, counted by A179009.
%Y A384347 The case of some proper choice is A384321, strict A384322, count A384317, strict A384318.
%Y A384347 These partitions are counted by A384323, strict A384319.
%Y A384347 A055396 gives least prime index, greatest A061395.
%Y A384347 A056239 adds up prime indices, row sums of A112798.
%Y A384347 A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
%Y A384347 A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
%Y A384347 A357982 counts strict partitions of prime indices, non-strict A299200.
%Y A384347 Cf. A048767, A382525, A382771, A382857, A383533, A383708, A384320.
%K A384347 nonn,more
%O A384347 1,1
%A A384347 _Gus Wiseman_, May 27 2025