cp's OEIS Frontend

A381431 Heinz number of the section-sum partition of the prime indices of n.

%I A381431 #10 Feb 27 2025 22:58:04
%S A381431 1,2,3,4,5,5,7,8,9,7,11,10,13,11,11,16,17,15,19,14,13,13,23,20,25,17,
%T A381431 27,22,29,13,31,32,17,19,17,25,37,23,19,28,41,17,43,26,33,29,47,40,49,
%U A381431 35,23,34,53,45,19,44,29,31,59,26,61,37,39,64,23,19,67,38
%N A381431 Heinz number of the section-sum partition of the prime indices of n.
%C A381431 The image first differs from A320340, A364347, A350838 in containing a(150) = 65.
%C A381431 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 A381431 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 A381431 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 A381431 The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.
%F A381431 A122111(a(n)) = A048767(n).
%e A381431 Prime indices of 180 are (3,2,2,1,1), with section-sum partition (6,3), so a(180) = 65.
%e A381431 The terms together with their prime indices begin:
%e A381431    1: {}
%e A381431    2: {1}
%e A381431    3: {2}
%e A381431    4: {1,1}
%e A381431    5: {3}
%e A381431    5: {3}
%e A381431    7: {4}
%e A381431    8: {1,1,1}
%e A381431    9: {2,2}
%e A381431    7: {4}
%e A381431   11: {5}
%e A381431   10: {1,3}
%e A381431   13: {6}
%e A381431   11: {5}
%e A381431   11: {5}
%e A381431   16: {1,1,1,1}
%t A381431 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A381431 egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
%t A381431 Table[Times@@Prime/@egs[prix[n]],{n,100}]
%Y A381431 The conjugate is A048767, union A351294, complement A351295, fix A048768 (count A217605).
%Y A381431 Taking length instead of sum in the definition gives A238745, conjugate A181819.
%Y A381431 Partitions of this type are counted by A239455, complement A351293.
%Y A381431 The union is A381432, complement A381433.
%Y A381431 Values appearing only once are A381434, more than once A381435.
%Y A381431 These are the Heinz numbers of rows of A381436, conjugate A381440.
%Y A381431 Greatest prime index of each term is A381437, counted by A381438.
%Y A381431 A000040 lists the primes, differences A001223.
%Y A381431 A003963 gives product of prime indices.
%Y A381431 A055396 gives least prime index, greatest A061395.
%Y A381431 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A381431 A122111 represents conjugation in terms of Heinz numbers.
%Y A381431 Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
%Y A381431 Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
%Y A381431 Cf. A000720, A003557, A005117, A047966, A051903, A066328, A091602, A116861, A130091, A212166, A380955.
%K A381431 nonn
%O A381431 1,2
%A A381431 _Gus Wiseman_, Feb 26 2025