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Number of normal patterns contiguously matched by the prime indices of n in increasing or decreasing order, counting multiplicity.
15

%I #10 Jun 27 2020 09:07:43

%S 1,2,2,3,2,3,2,4,3,3,2,5,2,3,3,5,2,5,2,5,3,3,2,7,3,3,4,5,2,4,2,6,3,3,

%T 3,7,2,3,3,7,2,4,2,5,5,3,2,9,3,5,3,5,2,7,3,7,3,3,2,7,2,3,5,7,3,4,2,5,

%U 3,4,2,10,2,3,5,5,3,4,2,9,5,3,2,7,3,3,3

%N Number of normal patterns contiguously matched by the prime indices of n in increasing or decreasing order, counting multiplicity.

%C First differs from A181796 at a(180) = 9, A181796(180) = 10.

%C First differs from A335549 at a(90) = 7, A335549(90) = 8.

%C 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.

%C We define a (normal) pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to contiguously match a pattern P if there is a contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) contiguously matches (1,1,2) and (2,1,1) but not (2,1,2), (1,2,1), (1,2,2), or (2,2,1).

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>

%H Gus Wiseman, <a href="/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>

%e The a(n) patterns for n = 2, 30, 12, 60, 120, 540, 1500:

%e () () () () () () ()

%e (1) (1) (1) (1) (1) (1) (1)

%e (12) (11) (11) (11) (11) (11)

%e (123) (12) (12) (12) (12) (12)

%e (112) (112) (111) (111) (111)

%e (123) (112) (112) (112)

%e (1123) (123) (122) (122)

%e (1112) (1112) (123)

%e (1123) (1122) (1123)

%e (11123) (1222) (1222)

%e (11222) (1233)

%e (12223) (11233)

%e (112223) (12333)

%e (112333)

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];

%t Table[Length[Union[mstype/@ReplaceList[primeMS[n],{___,s___,___}:>{s}]]],{n,100}]

%Y The version for standard compositions is A335458.

%Y The not necessarily contiguous version is A335549.

%Y Patterns are counted by A000670 and ranked by A333217.

%Y A number's prime indices are given in the rows of A112798.

%Y Contiguous subsequences of standard compositions are A124771.

%Y Contiguous sub-partitions of prime indices are counted by A335519.

%Y Minimal avoided patterns of prime indices are counted by A335550.

%Y Patterns contiguously matched by partitions are counted by A335838.

%Y Cf. A000005, A056239, A056986, A108917, A124770, A181796, A269134, A333224, A334299, A335457, A335837.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jun 26 2020