A335519 Number of contiguous divisors of n.

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

Offset

1,2

Comments

A divisor of n is contiguous if its prime factors, counting multiplicity, are a contiguous subsequence of the prime factors of n. Explicitly, a divisor d|n is contiguous if n can be written as n = x * d * y where the least prime factor of d is at least the greatest prime factor of x, and the greatest prime factor of d is at most the least prime factor of y.

Links

Table of n, a(n) for n=1..85.

Wikipedia, Permutation pattern

Gus Wiseman, Sequences counting and encoding certain classes of multisets

Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Formula

a(n) = A325770(n) + 1.

Example

The a(84) = 10 distinct contiguous subsequences of (2,2,3,7) are (), (2), (3), (7), (2,2), (2,3), (3,7), (2,2,3), (2,3,7), (2,2,3,7), corresponding to the divisors 1, 2, 3, 7, 4, 6, 21, 12, 42, 84.

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Length[Union[ReplaceList[primeMS[n], {___, s___, ___}:>{s}]]], {n, 100}]

Crossrefs

The not necessarily contiguous version is A000005.

Each number's prime indices are given in the rows of A112798.

Contiguous subsequences of standard compositions are counted by A124771.

Minimal avoided patterns of prime indices are counted by A335550.

Patterns contiguously matched by partitions are counted by A335838.

Cf. A000670, A056239, A056986, A181796, A325770, A334299, A335457.

Sequence in context: A318412 A365208 A322986 * A167447 A134687 A184395

Adjacent sequences: A335516 A335517 A335518 * A335520 A335521 A335522

Keyword

nonn

Author

Gus Wiseman, Jun 26 2020

Status

approved