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A384322 Heinz numbers of strict integer partitions with more than one possible way to choose disjoint strict partitions of each part, i.e., strict partitions that can be properly refined.

%I A384322 #8 Jul 27 2025 18:03:11
%S A384322 5,7,11,13,17,19,21,22,23,26,29,31,33,34,35,37,38,39,41,43,46,47,51,
%T A384322 53,55,57,58,59,61,62,65,67,69,71,73,74,77,79,82,83,85,86,87,89,91,93,
%U A384322 94,95,97,101,102,103,106,107,109,111,113,114,115,118,119,122
%N A384322 Heinz numbers of strict integer partitions with more than one possible way to choose disjoint strict partitions of each part, i.e., strict partitions that can be properly refined.
%e A384322 The strict partition (7,2,1) with Heinz number 102 can be properly refined into (4,3,2,1), so 102 is in the sequence.
%e A384322 The terms together with their prime indices begin:
%e A384322      5: {3}      46: {1,9}      85: {3,7}
%e A384322      7: {4}      47: {15}       86: {1,14}
%e A384322     11: {5}      51: {2,7}      87: {2,10}
%e A384322     13: {6}      53: {16}       89: {24}
%e A384322     17: {7}      55: {3,5}      91: {4,6}
%e A384322     19: {8}      57: {2,8}      93: {2,11}
%e A384322     21: {2,4}    58: {1,10}     94: {1,15}
%e A384322     22: {1,5}    59: {17}       95: {3,8}
%e A384322     23: {9}      61: {18}       97: {25}
%e A384322     26: {1,6}    62: {1,11}    101: {26}
%e A384322     29: {10}     65: {3,6}     102: {1,2,7}
%e A384322     31: {11}     67: {19}      103: {27}
%e A384322     33: {2,5}    69: {2,9}     106: {1,16}
%e A384322     34: {1,7}    71: {20}      107: {28}
%e A384322     35: {3,4}    73: {21}      109: {29}
%e A384322     37: {12}     74: {1,12}    111: {2,12}
%e A384322     38: {1,8}    77: {4,5}     113: {30}
%e A384322     39: {2,6}    79: {22}      114: {1,2,8}
%e A384322     41: {13}     82: {1,13}    115: {3,9}
%e A384322     43: {14}     83: {23}      118: {1,17}
%t A384322 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A384322 pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y],UnsameQ@@#&];
%t A384322 Select[Range[100],UnsameQ@@prix[#]&&Length[pof[prix[#]]]>1&]
%Y A384322 The non-strict version for no choices appears to be A382912, count A383710, odd A383711.
%Y A384322 The non-strict version for > 0 choice appears to be A382913, count A383708, odd A383533.
%Y A384322 These are the squarefree positions of terms > 1 in A383706, see A357982, A299200.
%Y A384322 The case of a unique choice is A383707, counted by A179009.
%Y A384322 Partitions of this type are counted by A384318.
%Y A384322 This is the strict/squarefree case of A384321, counted by A384317.
%Y A384322 The case of a unique proper choice is A384390, counted by A384319, non-strict A384323.
%Y A384322 A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
%Y A384322 A055396 gives least prime index, greatest A061395.
%Y A384322 A056239 adds up prime indices, row sums of A112798.
%Y A384322 A239455 counts Look-and-Say partitions, ranks A351294 or A381432.
%Y A384322 A279790 and A279375 count ways to choose disjoint strict partitions of prime indices.
%Y A384322 A351293 counts non-Look-and-Say partitions, ranks A351295 or A381433.
%Y A384322 Cf. A098859, A122111, A130091, A317142, A381454, A382525, A384005, A384320, A384347.
%K A384322 nonn
%O A384322 1,1
%A A384322 _Gus Wiseman_, Jun 01 2025