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

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:
   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

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