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A384007 Heinz numbers of non Look-and-Say section-sum partitions.

%I A384007 #5 May 22 2025 17:06:03
%S A384007 10,14,15,22,26,33,34,35,38,39,46,51,55,57,58,62,65,69,74,77,82,85,86,
%T A384007 87,91,93,94,95,100,106,111,115,118,119,122,123,129,130,133,134,141,
%U A384007 142,143,145,146,155,158,159,161,166,170,177,178,182,183,185,187,190
%N A384007 Heinz numbers of non Look-and-Say section-sum partitions.
%C A384007 First differs from A383514 in lacking 1000.
%C A384007 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.
%C A384007 An integer partition is Look-and-Say iff it is possible to choose a disjoint family of strict partitions, one of each of its multiplicities. These are ranked by A351294.
%C A384007 An integer partition is section-sum iff it is possible to choose a disjoint family of strict partitions, one of each of its positive 0-appended differences. These are ranked by A381432.
%e A384007 The terms together with their prime indices begin:
%e A384007     10: {1,3}    57: {2,8}      94: {1,15}
%e A384007     14: {1,4}    58: {1,10}     95: {3,8}
%e A384007     15: {2,3}    62: {1,11}    100: {1,1,3,3}
%e A384007     22: {1,5}    65: {3,6}     106: {1,16}
%e A384007     26: {1,6}    69: {2,9}     111: {2,12}
%e A384007     33: {2,5}    74: {1,12}    115: {3,9}
%e A384007     34: {1,7}    77: {4,5}     118: {1,17}
%e A384007     35: {3,4}    82: {1,13}    119: {4,7}
%e A384007     38: {1,8}    85: {3,7}     122: {1,18}
%e A384007     39: {2,6}    86: {1,14}    123: {2,13}
%e A384007     46: {1,9}    87: {2,10}    129: {2,14}
%e A384007     51: {2,7}    91: {4,6}     130: {1,3,6}
%e A384007     55: {3,5}    93: {2,11}    133: {4,8}
%t A384007 disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
%t A384007 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A384007 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A384007 Select[Range[100],disjointFamilies[prix[#]]=={}&&disjointFamilies[conj[prix[#]]]!={}&]
%Y A384007 Ranking sequences are shown in parentheses below.
%Y A384007 These partitions are counted by A383509.
%Y A384007 Negating both properties gives (A383516).
%Y A384007 A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
%Y A384007 A055396 gives least prime index, greatest A061395.
%Y A384007 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A384007 A098859 counts Wilf partitions (A130091), conjugate (A383512).
%Y A384007 A122111 represents conjugation in terms of Heinz numbers.
%Y A384007 A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).
%Y A384007 A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
%Y A384007 A383508 counts partitions that are both Look-and-Say and section-sum (A383515).
%Y A384007 A383510 counts partitions that are neither Look-and-Say nor section-sum (A383517).
%Y A384007 A383511 counts partitions that are Look-and-Say and section-sum but not Wilf (A383518).
%Y A384007 Cf. A000720, A001223, A212166, A238745, A325368, A383514, A383520, A383531, A384006.
%K A384007 nonn
%O A384007 1,1
%A A384007 _Gus Wiseman_, May 19 2025