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

Author

Gus Wiseman, Jun 26 2020

Keywords

• nonn

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.

Examples

			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.
		

Links

• 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.

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.

Programs

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}]

Formula

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