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A335519 Number of contiguous divisors of n.

%I A335519 #11 Jun 27 2020 09:07:50
%S A335519 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,
%T A335519 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,
%U A335519 6,4,7,2,12,2,4,6,6,4,7,2,10,5,4,2,10,4
%N A335519 Number of contiguous divisors of n.
%C A335519 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.
%H A335519 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>
%H A335519 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vSX9dPMGJhxB8rOknCGvOs6PiyhupdWNpqLsnphdgU6MEVqFBnWugAXidDhwHeKqZe_YnUqYeGOXsOk/pub">Sequences counting and encoding certain classes of multisets</a>
%H A335519 Gus Wiseman, <a href="/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>
%F A335519 a(n) = A325770(n) + 1.
%e A335519 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.
%t A335519 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A335519 Table[Length[Union[ReplaceList[primeMS[n],{___,s___,___}:>{s}]]],{n,100}]
%Y A335519 The not necessarily contiguous version is A000005.
%Y A335519 Each number's prime indices are given in the rows of A112798.
%Y A335519 Contiguous subsequences of standard compositions are counted by A124771.
%Y A335519 Minimal avoided patterns of prime indices are counted by A335550.
%Y A335519 Patterns contiguously matched by partitions are counted by A335838.
%Y A335519 Cf. A000670, A056239, A056986, A181796, A325770, A334299, A335457.
%K A335519 nonn
%O A335519 1,2
%A A335519 _Gus Wiseman_, Jun 26 2020