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A384393 Heinz numbers of integer partitions with more than one proper way to choose disjoint strict partitions of each part.

%I A384393 #8 Jun 03 2025 08:43:43
%S A384393 11,13,17,19,23,25,29,31,34,37,38,41,43,46,47,49,51,53,55,57,58,59,61,
%T A384393 62,65,67,69,71,73,74,77,79,82,83,85,86,87,89,91,93,94,95,97,101,103,
%U A384393 106,107,109,111,113,115,118,119,121,122,123,127,129,131,133,134
%N A384393 Heinz numbers of integer partitions with more than one proper way to choose disjoint strict partitions of each part.
%C A384393 By "proper" we exclude the case of all singletons, which is disjoint when n is squarefree.
%C A384393 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 A384393 The prime indices of 275 are {3,3,5}, with a total of 2 proper choices: ((3),(2,1),(5)) and ((2,1),(3),(5)), so 275 is in the sequence.
%e A384393 The terms together with their prime indices begin:
%e A384393     11: {5}      51: {2,7}      82: {1,13}
%e A384393     13: {6}      53: {16}       83: {23}
%e A384393     17: {7}      55: {3,5}      85: {3,7}
%e A384393     19: {8}      57: {2,8}      86: {1,14}
%e A384393     23: {9}      58: {1,10}     87: {2,10}
%e A384393     25: {3,3}    59: {17}       89: {24}
%e A384393     29: {10}     61: {18}       91: {4,6}
%e A384393     31: {11}     62: {1,11}     93: {2,11}
%e A384393     34: {1,7}    65: {3,6}      94: {1,15}
%e A384393     37: {12}     67: {19}       95: {3,8}
%e A384393     38: {1,8}    69: {2,9}      97: {25}
%e A384393     41: {13}     71: {20}      101: {26}
%e A384393     43: {14}     73: {21}      103: {27}
%e A384393     46: {1,9}    74: {1,12}    106: {1,16}
%e A384393     47: {15}     77: {4,5}     107: {28}
%e A384393     49: {4,4}    79: {22}      109: {29}
%t A384393 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A384393 pofprop[y_]:=Select[DeleteCases[Join@@@Tuples[IntegerPartitions/@y],y],UnsameQ@@#&];
%t A384393 Select[Range[100],Length[pofprop[prix[#]]]>1&]
%Y A384393 Without "proper" we get A384321 (strict A384322), counted by A384317 (strict A384318).
%Y A384393 The case of no choices is A384349, counted by A384348.
%Y A384393 These are positions of terms > 1 in A384389.
%Y A384393 The case of a unique proper choice is A384390, counted by A384319.
%Y A384393 Partitions of this type are counted by A384395.
%Y A384393 A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
%Y A384393 A055396 gives least prime index, greatest A061395.
%Y A384393 A056239 adds up prime indices, row sums of A112798.
%Y A384393 A239455 counts Look-and-Say partitions, ranks A351294 or A381432.
%Y A384393 A279790 and A279375 count ways to choose disjoint strict partitions of prime indices.
%Y A384393 A351293 counts non-Look-and-Say partitions, ranks A351295 or A381433.
%Y A384393 Cf. A382912, A383710, A382913, A383708, A383533.
%Y A384393 Cf. A179009, A357982, A381454, A382525, A383706, A383707, A384320, A384323.
%K A384393 nonn
%O A384393 1,1
%A A384393 _Gus Wiseman_, Jun 02 2025