cp's OEIS Frontend

A381440 Irregular triangle read by rows where row k is the Look-and-Say partition of the prime indices of n.

%I A381440 #6 Feb 28 2025 23:12:04
%S A381440 1,1,1,2,1,1,1,1,1,1,1,1,1,1,3,2,2,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,
%T A381440 1,1,1,1,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,2,2,1,1,1,1,1,1,1,1,1,2,1,1,1,
%U A381440 1,1,1,1,1,1,1,1,1,1,1,1
%N A381440 Irregular triangle read by rows where row k is the Look-and-Say partition of the prime indices of n.
%C A381440 Row lengths are A066328.
%C A381440 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 A381440 The Look-and-Say partition of a multiset or partition y is obtained by interchanging parts with multiplicities. For example, starting with (3,2,2,1,1) we get (2,2,2,1,1,1), the multiset union of ((1,1,1),(2,2),(2)).
%C A381440 The conjugate of a Look-and-Say partition is a section-sum partition; see A381431, union A381432, count A239455.
%e A381440 The prime indices of 24 are (2,1,1,1), with Look-and-Say partition (3,1,1), so row 24 is (3,1,1).
%e A381440 The prime indices of 36 are (2,2,1,1), with Look-and-Say partition (2,2,2), so row 36 is (2,2,2).
%e A381440 Triangle begins:
%e A381440    1: (empty)
%e A381440    2: 1
%e A381440    3: 1 1
%e A381440    4: 2
%e A381440    5: 1 1 1
%e A381440    6: 1 1 1
%e A381440    7: 1 1 1 1
%e A381440    8: 3
%e A381440    9: 2 2
%e A381440   10: 1 1 1 1
%e A381440   11: 1 1 1 1 1
%e A381440   12: 2 1 1
%e A381440   13: 1 1 1 1 1 1
%e A381440   14: 1 1 1 1 1
%e A381440   15: 1 1 1 1 1
%e A381440   16: 4
%e A381440   17: 1 1 1 1 1 1 1
%e A381440   18: 2 2 1
%e A381440   19: 1 1 1 1 1 1 1 1
%t A381440 Table[Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>ConstantArray[k,PrimePi[p]]]]//Reverse,{n,30}]
%Y A381440 Heinz numbers are A048767 (union A351294, complement A351295, fixed A048768, A217605).
%Y A381440 First part in each row is A051903, conjugate A066328.
%Y A381440 Last part in each row is A051904, conjugate A381437 (counted by A381438).
%Y A381440 Row sums are A056239.
%Y A381440 Row lengths are A066328.
%Y A381440 Partitions of this type are counted by A239455, complement A351293.
%Y A381440 The conjugate is A381436, Heinz numbers A381431 (union A381432, complement A381433).
%Y A381440 Rows appearing only once have Heinz numbers A381540, more than once A381541.
%Y A381440 A000040 lists the primes.
%Y A381440 A003963 gives product of prime indices.
%Y A381440 A055396 gives least prime index, greatest A061395.
%Y A381440 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A381440 A122111 represents conjugation in terms of Heinz numbers.
%Y A381440 Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
%Y A381440 Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
%Y A381440 Cf. A000720, A003557, A047966, A071178, A091602, A116861, A130091, A181819, A212166, A238744, A380955.
%K A381440 nonn,tabf
%O A381440 1,4
%A A381440 _Gus Wiseman_, Feb 28 2025