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

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, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95

Offset

1,2

Comments

First differs from A371088 in having a(86) = 121.

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 choosable.

Conjecture: Also numbers whose weakly increasing prime indices y satisfy y(i) >= i for every index i.

Links

Table of n, a(n) for n=1..67.

Example

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
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.

Mathematica

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

Crossrefs

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

See also A276078, complement A276079.

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

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

These are all the positions of nonzero terms in A387111, complement A387134.

The complement is A387113.

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

For partitions instead of initial intervals we have A387576, complement A387577.

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, A387110, A383706.

Sequence in context: A352489 A370422 A371170 * A371088 A368110 A380692

Adjacent sequences: A387109 A387110 A387111 * A387113 A387114 A387115

Keyword

nonn

Author

Gus Wiseman, Aug 23 2025

Status

approved