cp's OEIS Frontend

A382525 Number of times n appears in A048767 (rank of Look-and-Say partition of prime indices). Number of ordered set partitions whose block-sums are the prime signature of n.

%I A382525 #16 Apr 07 2025 09:26:36
%S A382525 1,1,1,1,1,0,1,2,1,0,1,1,1,0,0,2,1,1,1,1,0,0,1,1,1,0,2,1,1,0,1,3,0,0,
%T A382525 0,0,1,0,0,1,1,0,1,1,1,0,1,1,1,1,0,1,1,1,0,1,0,0,1,0,1,0,1,4,0,0,1,1,
%U A382525 0,0,1,1,1,0,1,1,0,0,1,1,2,0,1,0,0,0,0
%N A382525 Number of times n appears in A048767 (rank of Look-and-Say partition of prime indices). Number of ordered set partitions whose block-sums are the prime signature of n.
%C A382525 The Look-and-Say partition of a multiset or partition y is obtained by interchanging parts with multiplicities. Hence, the multiplicity of k in the Look-and-Say partition of y is the sum of all parts that appear exactly k times. 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 A382525 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, sum A056239.
%C A382525 Also the number of ways to choose a set of disjoint strict integer partitions, one of each nonzero multiplicity in the prime factorization of n.
%F A382525 a(2^n) = A000009(n).
%F A382525 a(prime(n)) = 1.
%e A382525 The a(27) = 2 partitions with Look-and-Say partition (2,2,2) are: (3,3), (2,2,1,1).
%e A382525 The prime indices of 3456 are {1,1,1,1,1,1,1,2,2,2}, and the partitions with Look-and-Say partition (2,2,2,1,1,1,1,1,1,1) are:
%e A382525   (7,3,3)
%e A382525   (7,2,2,1,1)
%e A382525   (6,3,3,1)
%e A382525   (5,3,3,2)
%e A382525   (4,3,3,2,1)
%e A382525   (4,3,2,2,1,1)
%e A382525 so a(3456) = 6.
%t A382525 stp[y_]:=Select[Tuples[Select[IntegerPartitions[#],UnsameQ@@#&]&/@y],UnsameQ@@Join@@#&];
%t A382525 Table[Length[stp[Last/@FactorInteger[n]]],{n,100}]
%Y A382525 Positions of positive terms are A351294, conjugate A381432.
%Y A382525 Positions of 0 are A351295, conjugate A381433.
%Y A382525 Positions of 1 are A381540, conjugate A381434.
%Y A382525 Positions of terms > 1 are A381541, conjugate A381435.
%Y A382525 Positions of first appearances are A382775.
%Y A382525 A000670 counts ordered set partitions.
%Y A382525 A003963 gives product of prime indices.
%Y A382525 A055396 gives least prime index, greatest A061395.
%Y A382525 A056239 adds up prime indices, row sums of A112798.
%Y A382525 A122111 represents conjugation in terms of Heinz numbers.
%Y A382525 A239455 counts Look-and-Say partitions, complement A351293.
%Y A382525 A381436 lists the section-sum partition of prime indices, ranks A381431.
%Y A382525 A381440 lists the Look-and-Say partition of prime indices, ranks A048767.
%Y A382525 Cf. A003557, A047966, A048768, A050361, A051903, A051904, A066328, A071178, A116861, A130091, A217605, A239964.
%K A382525 nonn
%O A382525 1,8
%A A382525 _Gus Wiseman_, Apr 05 2025