cp's OEIS Frontend

A387111 Number of ways to choose a sequence of distinct positive integers, one in the initial interval of each prime index of n.

%I A387111 #11 Aug 21 2025 10:04:01
%S A387111 1,1,2,0,3,1,4,0,2,2,5,0,6,3,4,0,7,0,8,0,6,4,9,0,6,5,0,0,10,1,11,0,8,
%T A387111 6,9,0,12,7,10,0,13,2,14,0,2,8,15,0,12,2,12,0,16,0,12,0,14,9,17,0,18,
%U A387111 10,4,0,15,3,19,0,16,4,20,0,21,11,4,0,16,4,22
%N A387111 Number of ways to choose a sequence of distinct positive integers, one in the initial interval of each prime index of n.
%C A387111 The initial interval of a nonnegative integer x is the set {1,...,x}.
%C A387111 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 A387111 The axiom of choice says that, given any sequence of nonempty sets, it is possible to choose a sequence containing an element from each. In the strict version, the elements of this sequence must be distinct, meaning none is chosen more than once.
%e A387111 The prime indices of 75 are (2,3,3), with initial intervals ({1,2},{1,2,3},{1,2,3}), with choices (1,2,3), (1,3,2), (2,1,3), (2,3,1), so a(75) = 4.
%t A387111 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A387111 Table[Length[Select[Tuples[Range/@prix[n]],UnsameQ@@#&]],{n,100}]
%Y A387111 Allowing repeated partitions gives A003963.
%Y A387111 For constant instead of distinct we have A055396.
%Y A387111 For multiset systems see A355529, A355744, A367771, set systems A367901-A367905.
%Y A387111 For divisors we have A355739, zeros A355740, strict case of A355731.
%Y A387111 For prime factors we have A355741, prime powers A355742, weakly increasing A355745.
%Y A387111 For integer partitions we have A387110.
%Y A387111 Positions of nonzero terms are A387112 (choosable).
%Y A387111 Positions of 0 are A387134 (non-choosable).
%Y A387111 A001414 adds up distinct prime divisors, counted by A001221.
%Y A387111 A061395 gives greatest prime index.
%Y A387111 A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
%Y A387111 A120383 lists numbers divisible by all of their prime indices.
%Y A387111 A289509 lists numbers with relatively prime prime indices.
%Y A387111 A324850 lists numbers divisible by the product of their prime indices.
%Y A387111 Cf. A000720, A326841, A335433, A335448, A355733, A355737, A355747, A357980, A383706, A387120, A387133, A387135.
%K A387111 nonn,new
%O A387111 1,3
%A A387111 _Gus Wiseman_, Aug 18 2025