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A335549
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Number of normal patterns matched by the multiset of prime indices of n in weakly increasing order.
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12
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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, 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, 3, 4, 2, 10, 2, 3, 5, 5, 3, 4, 2, 9, 5, 3, 2, 7, 3, 3, 3
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OFFSET
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1,2
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COMMENTS
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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.
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 match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).
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LINKS
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EXAMPLE
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The Heinz number of (1,2,2,3) is 90 and it matches 8 patterns: (), (1), (11), (12), (112), (122), (123), (1223); so a(90) = 8.
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
mstype[q_]:=q/.Table[Union[q][[i]]->i, {i, Length[Union[q]]}];
Table[Length[Union[mstype/@Subsets[primeMS[n]]]], {n, 100}]
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CROSSREFS
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The version for standard compositions instead of prime indices is A335454.
Permutations of prime indices are counted by A008480.
Patterns matched by compositions of n are counted by A335456(n).
Minimal patterns avoided by a standard composition are counted by A335465.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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