cp's OEIS Frontend

A387112 Numbers with (strictly) choosable initial intervals of prime indices.

%I A387112 #12 Aug 26 2025 09:53:31
%S A387112 1,2,3,5,6,7,9,10,11,13,14,15,17,19,21,22,23,25,26,29,30,31,33,34,35,
%T A387112 37,38,39,41,42,43,45,46,47,49,50,51,53,55,57,58,59,61,62,63,65,66,67,
%U A387112 69,70,71,73,74,75,77,78,79,82,83,85,86,87,89,91,93,94,95
%N A387112 Numbers with (strictly) choosable initial intervals of prime indices.
%C A387112 First differs from A371088 in having a(86) = 121.
%C A387112 The initial interval of a nonnegative integer x is the set {1,...,x}.
%C A387112 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 A387112 We say that a set or sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1,2,3},{1},{1,3},{2}) is not.
%C A387112 This sequence lists all numbers k such that if the prime indices of k are (x1,x2,...,xz), then the sequence of sets (initial intervals) ({1,...,x1},{1,...,x2},...,{1,...,xz}) is choosable.
%e A387112 The prime indices of 85 are {3,7}, with initial intervals {{1,2,3},{1,2,3,4,5,6,7}}, which are choosable, so 85 is in the sequence
%e A387112 The prime indices of 90 are {1,2,2,3}, with initial intervals {{1},{1,2},{1,2},{1,2,3}}, which are not choosable, so 90 is not in the sequence.
%t A387112 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A387112 Select[Range[100],Select[Tuples[Range/@prix[#]],UnsameQ@@#&]!={}&]
%Y A387112 Partitions of this type are counted by A238873, complement A387118.
%Y A387112 For partitions instead of initial intervals we have A276078, complement A276079.
%Y A387112 For prime factors instead of initial intervals we have A368100, complement A355529.
%Y A387112 For divisors instead of initial intervals we have A368110, complement A355740.
%Y A387112 These are all the positions of nonzero terms in A387111, complement A387134.
%Y A387112 The complement is A387113.
%Y A387112 For strict partitions instead of initial intervals we have A387176, complement A387137.
%Y A387112 A061395 gives greatest prime index, least A055396.
%Y A387112 A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
%Y A387112 A120383 lists numbers divisible by all of their prime indices.
%Y A387112 A367902 counts choosable set-systems, complement A367903.
%Y A387112 A370582 counts sets with choosable prime factors, complement A370583.
%Y A387112 A370585 counts maximal subsets with choosable prime factors.
%Y A387112 Cf. A003963, A005703, A052335, A239312, A335433, A335448, A355739, A387110, A383706.
%K A387112 nonn,new
%O A387112 1,2
%A A387112 _Gus Wiseman_, Aug 23 2025