A374252 - OEIS

A374252

Irregular triangle read by rows where T(n,k) is the number of permutations of the prime factors of n with k runs.

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

2,7

COMMENTS

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.

LINKS

Table of n, a(n) for n=2..86.

EXAMPLE

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:

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

MATHEMATICA

prifacs[n_]:=If[n==1, {}, Flatten[ConstantArray@@@FactorInteger[n]]];

Table[Length[Select[Permutations[prifacs[n]], Length[Split[#]]==k&]], {n, 100}, {k, PrimeOmega[n]}]

CROSSREFS

Row-lengths are A001222.

Row-sums are A008480 (number of permutations of prime factors).

Column k = 1 is A069513.

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.

A124767 counts runs in standard compositions, anti-runs A333381.

A333755 counts compositions by number of runs.

A335433 lists separable numbers, complement A335448.

A374250 maximizes sum of run-compression, for indices A373956.

Cf. A026549, A027748, A046660, A126706, A151821, A246655.

Sequence in context: A015374 A164058 A328712 * A319368 A092410 A100204

Adjacent sequences: A374249 A374250 A374251 * A374253 A374254 A374255

KEYWORD

nonn,tabf

AUTHOR

Gus Wiseman, Jul 07 2024

STATUS

approved