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A381437 Last part of the section-sum partition of the prime indices of n.

%I A381437 #6 Mar 02 2025 08:02:07
%S A381437 0,1,2,1,3,3,4,1,2,4,5,1,6,5,5,1,7,2,8,1,6,6,9,1,3,7,2,1,10,6,11,1,7,
%T A381437 8,7,3,12,9,8,1,13,7,14,1,2,10,15,1,4,3,9,1,16,2,8,1,10,11,17,1,18,12,
%U A381437 2,1,9,8,19,1,11,8,20,1,21,13,3,1,9,9,22,1,2
%N A381437 Last part of the section-sum partition of the prime indices of n.
%C A381437 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A381437 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 A381437 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 A381437 The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.
%F A381437 a(n) = A055396(A381431(n)).
%e A381437 The prime indices of 972 are {1,1,2,2,2,2,2}, with section-sum partition (3,3,2,2,2), so a(972) = 2.
%t A381437 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A381437 egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
%t A381437 Table[If[n==1,0,Last[egs[prix[n]]]],{n,100}]
%Y A381437 Positions of first appearances are A008578.
%Y A381437 The length of this partition is A051903.
%Y A381437 The conjugate version is A051904.
%Y A381437 For first instead of last part we get A066328.
%Y A381437 These partitions are counted by A239455, complement A351293.
%Y A381437 Positions of 1 are A360013, complement A381439.
%Y A381437 This is the least prime index of A381431 (see A381432, A381433, A381434, A381435).
%Y A381437 This is the last part of row n of A381436 (see A381440, A048767, A351294, A351295).
%Y A381437 Counting partitions by this statistic gives A381438.
%Y A381437 A000040 lists the primes, differences A001223.
%Y A381437 A055396 gives least prime index, greatest A061395.
%Y A381437 A056239 adds up prime indices, row sums of A112798.
%Y A381437 A122111 represents conjugation in terms of Heinz numbers.
%Y A381437 Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
%Y A381437 Cf. A000720, A001221, A001222, A003557, A047966, A116861, A130091, A181819, A380955.
%K A381437 nonn
%O A381437 1,3
%A A381437 _Gus Wiseman_, Feb 28 2025