cp's OEIS Frontend

A381436 Irregular triangle read by rows where row k is the section-sum partition of the prime indices of n.

%I A381436 #6 Feb 28 2025 23:12:09
%S A381436 1,2,1,1,3,3,4,1,1,1,2,2,4,5,3,1,6,5,5,1,1,1,1,7,3,2,8,4,1,6,6,9,3,1,
%T A381436 1,3,3,7,2,2,2,5,1,10,6,11,1,1,1,1,1,7,8,7,3,3,12,9,8,4,1,1,13,7,14,6,
%U A381436 1,5,2,10,15,3,1,1,1,4,4,4,3,9,7,1,16,3,2,2
%N A381436 Irregular triangle read by rows where row k is the section-sum partition of the prime indices of n.
%C A381436 Row-lengths are A051903.
%C A381436 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 A381436 The section-sum partition 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 A381436 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 A381436 The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.
%e A381436 The prime indices of 24 are (2,1,1,1), with sections ((2,1),(1),(1)), so row 24 is (3,1,1).
%e A381436 Triangle begins:
%e A381436    1: (empty)
%e A381436    2: 1
%e A381436    3: 2
%e A381436    4: 1 1
%e A381436    5: 3
%e A381436    6: 3
%e A381436    7: 4
%e A381436    8: 1 1 1
%e A381436    9: 2 2
%e A381436   10: 4
%e A381436   11: 5
%e A381436   12: 3 1
%e A381436   13: 6
%e A381436   14: 5
%e A381436   15: 5
%e A381436   16: 1 1 1 1
%t A381436 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A381436 egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
%t A381436 Table[egs[prix[n]],{n,100}]
%Y A381436 Row-lengths are A051903.
%Y A381436 Row sums are A056239.
%Y A381436 First part in each row is A066328.
%Y A381436 Taking length instead of sum gives A238744, Heinz numbers A238745, conjugate A181819.
%Y A381436 Partitions of this type are counted by A239455, complement A351293.
%Y A381436 Heinz numbers are A381431 (union A381432, complement A381433, fixed A000961, A000005).
%Y A381436 Rows appearing only once have Heinz numbers A381434, more than once A381435.
%Y A381436 Last part in each row is A381437, counted by A381438.
%Y A381436 The conjugate is A381440, Heinz numbers A048767 (union A351294, complement A351295).
%Y A381436 A000040 lists the primes.
%Y A381436 A003963 gives product of prime indices.
%Y A381436 A055396 gives least prime index, greatest A061395.
%Y A381436 A056239 adds up prime indices, row sums of A112798.
%Y A381436 A122111 represents conjugation in terms of Heinz numbers.
%Y A381436 Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
%Y A381436 Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
%Y A381436 Cf. A000720, A001222, A003557, A005117, A047966, A116861, A130091, A380955.
%K A381436 nonn,tabf
%O A381436 1,2
%A A381436 _Gus Wiseman_, Feb 28 2025