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A381432 Heinz numbers of section-sum partitions. Union of A381431.

%I A381432 #7 Feb 28 2025 10:35:51
%S A381432 1,2,3,4,5,7,8,9,10,11,13,14,15,16,17,19,20,22,23,25,26,27,28,29,31,
%T A381432 32,33,34,35,37,38,39,40,41,43,44,45,46,47,49,50,51,52,53,55,56,57,58,
%U A381432 59,61,62,64,65,67,68,69,71,73,74,75,76,77,79,80,81,82,83
%N A381432 Heinz numbers of section-sum partitions. Union of A381431.
%C A381432 First differs from A320340, A364347, A350838 in containing 65.
%C A381432 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 A381432 The section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
%C A381432 Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
%C A381432 The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.
%e A381432 The terms together with their prime indices begin:
%e A381432     1: {}
%e A381432     2: {1}
%e A381432     3: {2}
%e A381432     4: {1,1}
%e A381432     5: {3}
%e A381432     7: {4}
%e A381432     8: {1,1,1}
%e A381432     9: {2,2}
%e A381432    10: {1,3}
%e A381432    11: {5}
%e A381432    13: {6}
%e A381432    14: {1,4}
%e A381432    15: {2,3}
%e A381432    16: {1,1,1,1}
%e A381432    17: {7}
%e A381432    19: {8}
%e A381432    20: {1,1,3}
%e A381432    22: {1,5}
%e A381432    23: {9}
%e A381432    25: {3,3}
%e A381432    26: {1,6}
%e A381432    27: {2,2,2}
%t A381432 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A381432 egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
%t A381432 Select[Range[100],MemberQ[Times@@Prime/@#&/@egs/@IntegerPartitions[Total[prix[#]]],#]&]
%Y A381432 Partitions of this type are counted by A239455, complement A351293.
%Y A381432 The conjugate is A351294, union of A048767 (parts A381440, fixed A048768, A217605).
%Y A381432 Union of A381431 (parts A381436).
%Y A381432 The complement is A381433, conjugate A351295.
%Y A381432 A000040 lists the primes, differences A001223.
%Y A381432 A055396 gives least prime index, greatest A061395.
%Y A381432 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A381432 A122111 represents conjugation in terms of Heinz numbers.
%Y A381432 Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
%Y A381432 Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
%Y A381432 Cf. A000720, A003557, A047966, A051903, A066328, A116861, A130091, A212166, A238745, A239964, A317081, A381437.
%K A381432 nonn
%O A381432 1,2
%A A381432 _Gus Wiseman_, Feb 27 2025