A356069 Number of divisors of n whose prime indices cover an interval of positive integers (A073491).

1, 2, 2, 3, 2, 4, 2, 4, 3, 3, 2, 6, 2, 3, 4, 5, 2, 6, 2, 4, 3, 3, 2, 8, 3, 3, 4, 4, 2, 7, 2, 6, 3, 3, 4, 9, 2, 3, 3, 5, 2, 5, 2, 4, 6, 3, 2, 10, 3, 4, 3, 4, 2, 8, 3, 5, 3, 3, 2, 10, 2, 3, 4, 7, 3, 5, 2, 4, 3, 5, 2, 12, 2, 3, 6, 4, 4, 5, 2, 6, 5, 3, 2, 7, 3, 3

Offset

1,2

Comments

First differs from A000005 at 10, 14, 20, 21, 22, ... = A307516.

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.

Links

Table of n, a(n) for n=1..86.

Example

The a(n) counted divisors of n = 1, 2, 4, 6, 12, 16, 24, 30, 36, 48, 72, 90:
  1   2   4   6  12  16  24  30  36  48  72  90
      1   2   3   6   8  12  15  18  24  36  45
          1   2   4   4   8   6  12  16  24  30
              1   3   2   6   5   9  12  18  18
                  2   1   4   3   6   8  12  15
                  1       3   2   4   6   9   9
                          2   1   3   4   8   6
                          1       2   3   6   5
                                  1   2   4   3
                                      1   3   2
                                          2   1
                                          1

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
nogapQ[m_]:=m=={}||Union[m]==Range[Min[m], Max[m]];
Table[Length[Select[Divisors[n], nogapQ[primeMS[#]]&]], {n, 100}]

Crossrefs

These divisors belong to A073491, a superset of A055932, complement A073492.

The initial case is A356224.

The complement in the initial case is counted by A356225.

A000005 counts divisors.

A001223 lists the prime gaps.

A056239 adds up prime indices, row sums of A112798, lengths A001222.

A328338 has third-largest divisor prime.

A356226 gives the lengths of maximal gapless intervals of prime indices.

Cf. A028334, A029709, A055874, A070824, A119313, A137921, A287170, A307516, A356223, A356233-A356237.

Sequence in context: A355698 A087990 A335037 * A179940 A138707 A358099

Adjacent sequences: A356066 A356067 A356068 * A356070 A356071 A356072

Keyword

nonn

Author

Gus Wiseman, Aug 28 2022

Status

approved