cp's OEIS Frontend

A387113 Numbers whose prime indices do not have (strictly) choosable initial intervals.

%I A387113 #7 Aug 26 2025 09:53:39
%S A387113 4,8,12,16,18,20,24,27,28,32,36,40,44,48,52,54,56,60,64,68,72,76,80,
%T A387113 81,84,88,90,92,96,100,104,108,112,116,120,124,126,128,132,135,136,
%U A387113 140,144,148,150,152,156,160,162,164,168,172,176,180,184,188,189,192
%N A387113 Numbers whose prime indices do not have (strictly) choosable initial intervals.
%C A387113 The initial interval of a nonnegative integer x is the set {1,...,x}.
%C A387113 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 A387113 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 A387113 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 not choosable.
%e A387113 The prime indices of 18 are {1,2,2}, with initial intervals ({1},{1,2},{1,2}), which have choices (1,1,1), (1,1,2), (1,2,1), (1,2,2), and since none of these are strict, 18 is in the sequence.
%e A387113 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 not the sequence.
%e A387113 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 in the sequence.
%e A387113 The terms together with their prime indices begin:
%e A387113     4: {1,1}
%e A387113     8: {1,1,1}
%e A387113    12: {1,1,2}
%e A387113    16: {1,1,1,1}
%e A387113    18: {1,2,2}
%e A387113    20: {1,1,3}
%e A387113    24: {1,1,1,2}
%e A387113    27: {2,2,2}
%e A387113    28: {1,1,4}
%e A387113    32: {1,1,1,1,1}
%e A387113    36: {1,1,2,2}
%e A387113    40: {1,1,1,3}
%e A387113    44: {1,1,5}
%e A387113    48: {1,1,1,1,2}
%e A387113    52: {1,1,6}
%e A387113    54: {1,2,2,2}
%e A387113    56: {1,1,1,4}
%e A387113    60: {1,1,2,3}
%e A387113    64: {1,1,1,1,1,1}
%t A387113 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A387113 Select[Range[100],Select[Tuples[Range/@prix[#]],UnsameQ@@#&]=={}&]
%Y A387113 For partitions instead of initial intervals we have A276079, complement A276078.
%Y A387113 For prime factors instead of initial intervals we have A355529, complement A368100.
%Y A387113 For divisors instead of initial intervals we have A355740, complement A368110.
%Y A387113 These are the positions of 0 in A387111, complement A387134.
%Y A387113 The complement is A387112.
%Y A387113 Partitions of this type are counted by A387118, complement A238873.
%Y A387113 For strict partitions instead of initial intervals we have A387137, complement A387176.
%Y A387113 A061395 gives greatest prime index, least A055396.
%Y A387113 A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
%Y A387113 A120383 lists numbers divisible by all of their prime indices.
%Y A387113 A367902 counts choosable set-systems, complement A367903.
%Y A387113 A370582 counts sets with choosable prime factors, complement A370583.
%Y A387113 A370585 counts maximal subsets with choosable prime factors.
%Y A387113 Cf. A003963, A005703, A052335, A239312, A335433, A335448, A355739, A355747, A370592, A383706, A387110.
%K A387113 nonn,new
%O A387113 1,1
%A A387113 _Gus Wiseman_, Aug 24 2025