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

4, 8, 12, 16, 18, 20, 24, 27, 28, 32, 36, 40, 44, 48, 52, 54, 56, 60, 64, 68, 72, 76, 80, 81, 84, 88, 90, 92, 96, 100, 104, 108, 112, 116, 120, 124, 126, 128, 132, 135, 136, 140, 144, 148, 150, 152, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188, 189, 192

Offset: 1

Author

Gus Wiseman, Aug 24 2025

Keywords

• nonn
• new

Comments

The initial interval of a nonnegative integer x is the set {1,...,x}.

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.

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.

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.

Examples

			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.
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.
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.
The terms together with their prime indices begin:
    4: {1,1}
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   27: {2,2,2}
   28: {1,1,4}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   44: {1,1,5}
   48: {1,1,1,1,2}
   52: {1,1,6}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}
		

Crossrefs

For partitions instead of initial intervals we have A276079, complement A276078.

For prime factors instead of initial intervals we have A355529, complement A368100.

For divisors instead of initial intervals we have A355740, complement A368110.

These are the positions of 0 in A387111, complement A387134.

The complement is A387112.

Partitions of this type are counted by A387118, complement A238873.

For strict partitions instead of initial intervals we have A387137, complement A387176.

A061395 gives greatest prime index, least A055396.

A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.

A120383 lists numbers divisible by all of their prime indices.

A367902 counts choosable set-systems, complement A367903.

A370582 counts sets with choosable prime factors, complement A370583.

A370585 counts maximal subsets with choosable prime factors.

Cf. A003963, A005703, A052335, A239312, A335433, A335448, A355739, A355747, A370592, A383706, A387110.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Select[Tuples[Range/@prix[#]],UnsameQ@@#&]=={}&]