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A374252
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Irregular triangle read by rows where T(n,k) is the number of permutations of the prime factors of n with k runs.
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6
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1, 1, 1, 0, 1, 0, 2, 1, 1, 0, 0, 1, 0, 0, 2, 1, 0, 2, 1, 1, 0, 2, 0, 2, 1, 0, 0, 0, 1, 0, 2, 1, 1, 0, 2, 1, 0, 2, 0, 2, 1, 0, 2, 2, 0, 1, 0, 0, 2, 1, 0, 0, 0, 2, 1, 1, 0, 0, 6, 1, 1, 0, 0, 0, 0, 0, 2, 0, 2, 0, 2, 0, 2, 2, 2, 1, 0, 2, 0, 2, 0, 2, 2, 0, 1
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OFFSET
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2,7
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COMMENTS
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An alternative form of this sequence (with the same data) has offset 1 and begins with an empty row.
Note that the prime factors of n are separable (A335433) iff the last term of row n is positive.
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LINKS
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EXAMPLE
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The T(36,3) = 2 permutations are (2,3,3,2) and (3,2,2,3).
Row n = 72 counts the following permutations:
. (2,2,2,3,3) (2,2,3,3,2) (2,2,3,2,3) (2,3,2,3,2)
(3,3,2,2,2) (2,3,3,2,2) (2,3,2,2,3)
(3,2,2,2,3) (3,2,2,3,2)
(3,2,3,2,2)
Triangle begins:
1:
2: 1
3: 1
4: 1 0
5: 1
6: 0 2
7: 1
8: 1 0 0
9: 1 0
10: 0 2
11: 1
12: 0 2 1
13: 1
14: 0 2
15: 0 2
16: 1 0 0 0
17: 1
18: 0 2 1
19: 1
20: 0 2 1
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MATHEMATICA
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prifacs[n_]:=If[n==1, {}, Flatten[ConstantArray@@@FactorInteger[n]]];
Table[Length[Select[Permutations[prifacs[n]], Length[Split[#]]==k&]], {n, 100}, {k, PrimeOmega[n]}]
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CROSSREFS
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Row-sums are A008480 (number of permutations of prime factors).
For compositions instead of permutations of prime factors we have A238130.
Last column is A335452 (where k = A001222(n)), which counts separations.
Position of the last positive term in row n is A373957(n).
The number of zeros at the end of row n is A374246(n).
The number of nonzero terms in row n is A374247(n).
A001221 counts distinct prime factors.
A003242 counts run-compressed compositions, i.e., anti-runs.
A333755 counts compositions by number of runs.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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